Interpolation spaces and i,~terpolation methods (9.
by ~N. A.RONSZAJN AND E. GAGLIAI~DO
~m.mary. - (See iv, troductio~).
Introduction.- The beginning of what we now call interpolation methods between BANAC~ spaces was the convexity theorem of )][. RIESZ [13] in the late 1920's. This theorem gives an interpolation metlmd for couples [LP(di~), Lq(d~t)]. In the late 1930's, MAlCCINKIEWleZ [11] obtained an extension of the M. RIESz interpolation method to couples formed by weak LP-spaces. In the 1950's E. )I. S~EIN and G. WEISS [14] extended the method further by admitting couples [Lv(d~), Lq(dv)] with different measures l~ and v. Until the late 1950's, the research in interpolation methods between sp~ees remained essentially within the frame of couples of L ~ spaces. At the end of 1958, J.M. LIOZ,~s gave the first proof of the inter- polation theorem for quadratic interpolation between HILBER~' spaces t2). This work gave impetus to the research on interpolation methods for arbitrary couples of BA~c~tOK spaces. Since then several authors have introduced and developed a number of different interpolation methods for couples of general B,t~ACI-I spaces. We mention here, for example, the methods of E. GAeLIARDO [3], [4], [5], ft. L. LIONS, [6], [7], [8], [9], A. CALDERON [2], ~. PEETRE [t2], and S.G. KREIN [10]. [n the presence of so many different interpolation methods it seemed timely to study the general structure of alt possible methods: to determine all of them and to analyze the properties which are common to all. This is essentially the subject of the present paper. The research was prompted by the following specific consideration.
(t} Research done under 5TSF Grant G17057 snd OSR Contract N. 583 (!3). i~') Note by N. Aron.szajn. I take this occasion to correct a misunderstanding which seems to have arisen in connection with references to my work in the first paper of J. L. Ltoxs ([6]). Althoagh foL~ some time [ was aware of the importaace of quadratic interpolation in the theory of B~SSEr~ potentials and in their applications (see [1l), I did n ~t have the proof of the relevant interpolation theorem, In the spring of 1958 I told J.L. LIoNs about the problem~ and in the fall of 1958 he sent me the proof. 52 N. ARo~-szAJ~ - E. GA(;L~O: lnterpotatio~ spaoes, etc.
For some applications it becomes apparent that it is not always of importance to develope a specific method, but rather to determine if a general interpolation method (~) exists which assigns to a given couple of B~Ac~ spaces a given intermediate space. In other words, the problem is to characterize those intermediate spaces between two given BANAC~ spaces for which such a general interpolation method exists. A solution to this problem is given in Corollary [14. VIIi. The relevant intermediate spaces are those which we call (~interpolation spaces ~ between two given BA~Ac~ spaces. This result led to a thorough study of interpo. lation spaces which comprises the greater part of the paper and is presented in Chapter II. A number of unexpected facts concerning interpolation spaces were discovered and will be indicated in the brief summary of the paper which follows. The first three sections of Chapter I-Preliminaries, contains material familiar to all who have worked in the domain, but it is collected here in order to"give a precise and intrinsic definition of the notion of <
Chapter I[ deals with interpolation spaces. In § 5 we introduce the normalized BANAC]/[ subspaces of a given B~_NAc~ space. The main theorem is that the lattice of these subspaees is complete and we give the construction of the joint and meet for an arbitrary class of such subspaces. In § 6 we introduce the intermediate and interpolation spaces, their normalizations and the B~_~'~Acl~ algebra ~;[V, W] for a compatible couple of BA~ACK spaces V and W; ~;[V, ~¥] is composed of all linear operators on V + W transforming boundedly V into V and W into W.
(3) We say that a method is "general" if it is defined for all compatible couples of BA~ACH spaces (or~ if we want to avoict any possible antinomies in se~ theory we should rather say ~defined on arbitrarily large classes of couples of compatible B~NACg spacos). g. ARoyszAa~ - E. Gh(~'LL~m)O: Interpolations. ,spaces, etc. 53
§ 7 deals with the case when V N W is not dense in V or W. We describe the restrictions which this hypothesis imposes on the interpolation spaces.
In § 8 we consider the ease when the conjugate spaces of V and W form, in a natural way, a compatible couple. This is the case when V G W is dense both in V and W. We then study the relationship between the interpolation spaces (~f IV, W] and [V*~ W*]. The adjoints of operators in ¢g[V, W] form a suba|gebra ~* of ~[F*, W*]. We obtain the rather unexpected result that ~;*~ ~[V*, W*] if and only if VN W is reflexive.
In § 9 we consider the properties of, and the relationships between the BANAC]t algebras ~[V, W], ~A (this is the algebra of bounded linear opera- tots on an interpolation space A) and ~;A[V, W] (the subalgebra of ¢g[V, W] which is the closure of the set of operators such that T(V+ WtcA}.
In § 10 we consider for a B~.NAc~ subspace A of a BA~'AC~ space E, the space A~E)- the completion o[ A tel. E. (This notion was illtroduced by E. GAGL][ARDO [4]). We give a formula iTheorem [10. V]) expressing A-(I~') in terms of the conjugate spaces of A and E. The main source of appli. cations o[ the notion of relative completion to the stud~ ~ of interpolation spaces is Lemma [10. X]. By using this lemma we obtain the following- again, rather unexpected result: whereas V-J- ~, and VN W are always interpolation spaces between Vand W, V and W, with some trivial exceptions, are never interpolation space.s between V-~ W and V N IV. This result indicates t~aat it is not an ~unesseutiaI>> restriction when one considers only couples IV, W] with WC V.
In § 11 we come back to interpolation spaces. We construct the minimal interpolation spaces containing a given intermediate space. In par- ticular, for any element ue V-{- W we construct tile minimal interpolation. space ~;(u) containing u: ~(u) is formed by all elements Tu for Te~[V, W] For u :~0, V N W is always contained in ~(u} but never dense in it {unless ueVN W when VA W--~;(u)). By using a normalized version ~,~ of ~;(~) we can construct all interpolation spaces between V and W (see Theorem [11. VII).
In § ][2 we analzze the structure of the minimal subspaces ~(u). For u ~ V N I,V, ~(u) always contains [u]--[- V N W, i.e., the subspace generated by V N W and u. The last statement in this section is to the effect that ~(u) -- [u] -{- VN W only in ver Z exceptional cases, namely when u~(vn W)(V+w). That this case can actually arise is shown in an example constructed in the last Remark of the section. 54 :N. ARONSZAJN - E. GAGL:[ARDO: Interpolatio~ spaces, etc.
Chapter III deals with our main results concerning interpolation me- thods and interpolation theorems.
In §13 we consider triples IV, W; A]i.e., compatible couples of BA~Ac~ spaces fIT, W] with an intermediate space A between them, and study the unilateral or bilateral interpolation theorems between two triples. The notions and exact formulas needed for the main theorem in § 14 are introduced. In addition, we consider the question of optimal interpolation theorems between two triples and obtain, somewhat unexpectedly, that in any such optimal interpolation theorem the intermediate spaces in each triple must be, actually, interpolation spaces.
In § 14 we pass to interpolation methods on a class of compatible BANACH couples. We define uniform and normalized interpolation methods. Each uniform method can be normalized. There are no known examples of methods which cannot be rendered uniform. An interpolation method assigns to each couple of B¢r~cAc~: spaces an intermediate space. This intermediate space must be an interpolation space. Corollary [14. VII] gives the result mentioned previously in this introduction, that for a given interpolation space A between V and W (A can be assumed normalized) there exist general interpolation methods (normalized} which assign A to the couple [V, W]. The more general Main Theorem [14. VI] states that if we have any two classes J{Cg~ of compatible BA~AcI~ couples, then any normalized method defined on J{ can be extended to a normalized method defined on ~, and among all these extensions there are two extreme ones. The original iutention of the authors was to include a study of conju- gate and self-conjugate interpolation methods, and also a study of existing interpolation methods with reference to the results of this paper. However, the present work has already greatly exceeded its expected length and we had to abandon these plans.
CnAPTrm [.
Prelimin~ries,
§ 1. - Linear c.uples.- We say that W is linearly contained in V (or V linearly contains W): W C V (or V D W t l Z if V and W are vector spaces, I~7 is a subspace of V and the identity mapping of W into V is linear. N. Aao~sz.~a~ - E. GAGLIARDO: Interpolation spaces, etc. 55
We will use the convention that a subspace is never empty (contains at least the origin) and hence a linear mapping always has a non-empty domain. For any two abstrac~ sets A and B, their intersection, A n B, deter. mines the identification mapping from A to B. Consider VcE and WcE; they obviously satisfy the following property :
(l.1)- The identification mapping z front V into W is a linear isomer. phism.
[1.I] DEFINITIO:N. - Two vector spaces V and W form a linear couple [V, W] if the relation {l.1) holds. Let IV, W] and IV1, W1] be linear couples and S a mapping of VU W into V1 t2 W1 which transforms linearly V into 171 and W into W~. We say then that S is a linear mapping of [V, W] into [V, I¥~]. If the mapping S is one-one and on|o V~ U W1 we say that it is an isomorphism. The inverse mapping is ti~ea also an isomorphism and the two couples are called isomorphic (by S). If V~C E and IiV~CE, then an isomorphism S of [17, WJ onto [V,, ~z~] is called an embedding into E and [I7, W] is then embedded (by SI into E. Consider a linear couple [17.. W] and the direct sum, V+ ~5 Define the subspace Z of the direct sum as follows:
(i.2) z = [{v, = -
where z is the identification mapping from V into W. Then the mapping defined on V and W by
(L3) v {v, o)+z, u,}+z
is obviously an embedding of [V, W] into the quotient space (Vq-W)/Z. In the remainder of the paper we will always identify the elements of V or W with their images (1.3} and therefore we will write
(1.4j (v-i- w)/z= v+ w 56 ~N. ARONSZAJI~ ~ - E. GAGLIAICDO: D~terpolatio~:~ spaces, etc.
If VcE and WCE and we form Che sum of V and Win E--tempo- rarily denoted tV & W)z-- then thel:e is. an obvi,)us canonical extension of the identity mapping of V/_) W into E to an-isomorphism of V+ W onto IV+ W)E which allows us to identify canonically IV+ W}~ with V+ W. In the following therefore IV+ W}E will be denoted by V + 1/V. The proof of the following statement is obvious.
[1.Ii] - If S is a linear mapping of IV, W] into [V~, W~] then it has a unique linear extension to a mapping of V+ W into V~-~-W~. If S is an isomorphism onlo. then the extension becomes an isomorfism ol V+ W onto V~+ W~. If, in addition, S is an embedding into E, then the exten- sion becomes an isomorphism of V+ W into E.
Consider two arbitrary vector spaces V and W. Their identification mapping is not necessarily linear (its domain may even be empty). Some- times it is possible to extend the identification mapping to a linear isomor- phism x from V into W. We can then idelitify the elements of V and W such that +~:-~ xv. We thus obtain h'om V and W a linear couple [I/~, W,] with identification mapping ~, defined as follows: in V+ W consider the subspace ),~ of elements {v,--xv}; then the canonical isomorphisms v--: (v, 0} + Z:, w--+ Iv, 0} + Z:, transform V and W o~lto subspaces V+ and W+ of tV~- W}/Z~. A linear couple IV', W+] is called an extension of a linear couple {V, W] if VC V' and W C W'. In terms of the identificatior~ mappings 1 l and ~' of the two couples we have the following obvious statement.
[l.III] - In order that the four vector spaces V, W, V', W' form linear couples lIT. W] and [g'~ W] s~eh that the second be an extension of the first, it is necessary and sufficient that V C V', W C W', that ~ and x' be linear isomorphisms and ~' be an exte~sion of x outside of [i f, W] (i.e., if w:~'v, then either w--~v or v~ V and sv~ W).
§ 2. Compatible couples. - Let E be aH:USDORFF vector spaee and V a subspace of E with its own topology (not necessarily the one induced by E}. We write V'CE c if the identity mapping V-~ E is continuous. We will say then that V is continuously eontai~ed in E ,or that V is a topological subspace of E. If ~-. fl~.RONSZAJN - E. GAGLIARDO: Interpolation 8paees~ etc. 57
VCE and WCE, we consider on VN W and V-I- W canonical tope- e logics defined as follows: 10. on V(~ IV we take the coarsest topology among those Which are finer than the ones induced by V and W; 20. on ~'--k W we take the finest among those which induce on V and W topo- logies coarser than the one of V and W respectively. An immediate argument shows that with these topologies Vfq WCE and Vq- WCE. We will be mostly interested in normed topologies. If the norms on V and W are II V]lv and II ~v ilw, the canonical topologies on V('t W and Vq-W are given by
('24) liuilvnw=,nax[i:ulJv, lluijw ], Iluljv÷w-- inf (ji.vjjr-Fllwjiw)(').
If Y and W are normed spaces and VCE, and WCE~ then obviously c c we have
(2.2) lu. IC VN W~ Ilu,,--vllv~0, Ilu.-~vlw~O=>v-wevn W.
This property has a meaning for an arbitrary linear couple of normed vector spaces and leads to the definition
[2.I] - DEFI~SITIO~Z. - A linear couple IV, W] of normed vector spaces is calted a compatible couple if property (2.2)holds. Let [V. H'] and [V1, W1] be two linear couples of HAUSDOI~FF vector spaces, and let S be a linear m:~pping of the first into the second. If S maps continuously V into V~ and W into W~ then we say that S is a continuous mapping of IV, W] into [V~, H,q]. If S is an embedding of [V, W] into E (with E a HAUSDOm~Y vector space) which maps continuously V and W into E, we say that S is a continuous embedding. In this case if we transfer the topologies of V and W by the isomorphism S onto S(V) and S(W) respectively, we obviously get S(V) CE and S(W) CE. One proves immediately, by using the preceding considerations, the following theorem :
(4) Other (equivalent} norms are sometimes preferable. For instance, when the norms in V and W are quadratic, the following norms on V fl W and V-t- t} are also quadratic:
::u ~nw_[fiu Ul!v_~,W = inf [tlv,lv+li'vvI ~' A,w~ 1112 '0 -~-T/,~---*'bl~
Annati d4. Ma~ematiea 8 58 .N~. ARONSZAJN - E. GAGLIARD0: I~terpolatioJ~ spaces~ etc.
[2.II].- TI~EO]~]~)I.- a) Let IV, W] be a compatible couple. Then the canonical embedding (1.3)of IV, W] into (V-~ W)/Z (with norm (2.1) on V-~ W) is continuous. b) Let the normed linear couple [I/, W] be continuously embedded into the Hausdor]7 vector space E. Then [V, W] is a compatible couple and the induced isomorphism of V ~- W into E is continuous. [2.II1]. REMARK. - If IV, W] is a compatible couple and W-- VA WC Y it does not follow, in general, that WC V if. i. V = L=(R), W-- Lo~(R) -- the o, class of functions in Lm(lg) vanishing outside of a compact). However, if we change the norm on V to the norm {2.1), the relation WC V follows from the c above theorem. [2.I¥]. RE:~IARK - If [V, TkV] is a compatible couple of Banach spaces and W-- VA WC V then W C V ~by immediate application of closed c graph theorem). i2.V]. REHAI~K. - It is welt known and easily proved that if [17, W] is a compatible couple of BA~ACH spaces, then V A W and Vq-W with the norms (2.1} are BA~cl-I spaces.
[2.VI] REM:ARK;- The compatibility of a linear couple of normed spaces is clearly equivalent to the fact that the identificatiou mapping is closed.
§ 3. Completely compatible couples. -Before passing to the main considerations of this section, we make a few remarks. A completion V of a normed space V is a BA~AC~ space containing V as a dense subspace and such that LIvI]ff:llv :v for ve V. The classical abstract completion ~a (formed by equivalence classes of CAUCI~¥ Sequences in V) is not the only useful one. if V is a subspace of a HAUSDORF~ vector space E, there exists sometimes a completion of V in E, that is, a completion which is C E. If a completion in E exists, it is unique and will be denoted by I~E. Any two completions are canonically isometrically isomorphic. The question of existence of l~E is solved by the following:
[3.1] TI{EOREM.- ]If a normed space V is a subspace of a Hausdorffvector space E, then ~7E exists if and only if: a) V cE: b) every Cauchy sequence in V converges in E; c) when a Cauchy sequence in V converges in E to O, it also converges in V to O. N. ARo~sz.~J~ - E. GAGLt2*RDO: Interpolation, spaces, etc. 59
The proof is elementary but not very short. We omit it since our interest will be centered in the case when E is a BANAC~[ space and this case will be settled as a simple corollary of our forthcoming theorem.
[3.I[]. RE~AR]~. - If E is a BA~AC~ space, the condition b) in the above theorem is clearly implied by a). However c) is not implied by a) (example: on the interval l--[--1-- [3.III] REMARK. - ~f V and W are subspaces of a HAUSDORFF vector space E and ~E and ~V~ exist, then [~z, I~VEl is a completion of IV, PV]. [3.IV] REMARK.- Two different completions are not necessarily iso. morphic: if they are denoted by [l~, 17V] and [~', ~7,], then the canonical isomorphisms F,(-~ V' and I~T( ~ 17V' which leave respectively V and W invariant do not transform necessarily ~7 (~ ~-TV onto ~7'A I7V'. Suppose that there exists a completion [~ VV] of the compatible couple IV, W]. By the canonical isomorphisms V<-> YV", W<-> I/V~ we then transfer the identification mapping ~: of IV, ~V] to ~'Va and ~V¢a and obtain thus an identification mapping z' and a linear couple [V~,,~~' W~,]"~' which is a completion of [V, W]. Since z' is an extension of the identification mapping -: of [I/, W] and z' is a closed mapping from ]~a to ~d a, the mapping -~ must be closable in I 7" and -I/~da. A direct translation of the last fact is the following property: (8.1) If }u~}C V(h V~ and lu,~} is Cauchy in V and W~ then i[ u, !tv~ 0 is equivalent to l] u~ ttw--~ O. [3.V] DEHlgITIO1,;.- A. compatible~coaple lIT, W] is called completely compatible (v-compatible) if propert, y (3.1) holds. [3.VI] REIVIARK.- Obviously, for a linear couple of BA~xc~ spaces compatibility is equivalent to complete compatibility. [3.VII] TI-IEOREM. - Let IV, W] be a compatible couple and ~: its identi- fication mapping, a} All the completions of [V, W], up to isomorphisms, are given by [V~,, I~,] where is any closed identification mapping from V~ 60 ~. AR0:NSZASN - E. GAGLIARDO: [~te~'poIatio~$ spaces, etc. to W ~ which is an extension of z outside of [V, W]. b) Such mappings ~' exist if and only if [V, W] is completely compatible, c) If [V, W] is c-compatible, then among all ~' satisfying the conditions of a) there exists a minimal xo-the closure of x in I?~ and ~; the identification ~o can be directly described as follows: (3.2) For v~ V~. we ~VVa, ~°v= w means that for some sequence ( u,} C V (~ W which is Cauchy in V and W, Ilu,,~vi] ff'~O and [I u,~-- w I1¢¢.~ O. The proof of this theorem results immediately from the preceding consi- derations. We note that the extension xo of x is outside of [V~ W] since is closed in V and W. [3.VIII]. DEFInitIOn. - For a completely compatible couple [17, W] the completion [V¢o, VtT~0] where z ° is described by (3.2) will be called the minimal completion of IV, W]. [3.IX]. COROLLARY. - Let V be a normed space and E a Banach space such that V C E. The completion of V in E exists if and only if the linear couple [V~ E] ~is c-compatible. PROOF. - If ITE exists, then [l~E, El is a completion of [17, Ej; hence [V~ El is c-compatible. On the other hand, if IV, El is c-compatible, we ~Cb consider the minimal completion [V~o, E~o] of [V, El. Since VcE, the c identification mapping z of IV, E~ is a continuous mapping of V in E. Its closure ~o in ~'a and E therefore maps ~(a continuously in E, hence identifies l~" with a subspace of E which is ~E. [3.X]. RE~ARK. - In some previous papers the first author introduced a notion similar to complete compatibility. The notion was called (( compatibility z and was introduced for two norms defined on the same vector space. We had then two, iu general different, normed vector spaces forming a linear couple IV, WJ with V: V N W ~ W. The norms were called compatible if the condition (3.i) was satisfied. It is immediately verified that in the present ease~ condition (3.i) alone is equivalent to com- plete compatibility of [V~ W]. § 4. Changing compatible couples into completely compatible. - Let us suppose now that the couple [V, W] is compatible, but not necessarily completely compatible. Our problem will be to change the norms in V, W, finding two new norms (not necessarily equivalent to the previous) in order N. ARONSZAJN - E. GAGLIARD0: Interpolation spaces, etc. 61 to have a completely compatible couple. This can be done in many different ways, but among all possible ways there is a canonical one with special properties: [4.I]. THEOREM. - Let IV, W] be compatible. Let ~ = { II II ~, l[ II ~v} be the class of all norms on V and W which make IV, W] completely compa. tibIe and such that (4.1) /I v I1~ II v i1~, II w ]]~ II w tlw, The class C is not empty and has a maximal element; namely, if we define (with v,~ and w,~ ~ V N W, n -- 1> 2, ...) / 1!v1111= inf lira IIv--v,,l]v for all Iv,} Cauchy in V and II v,, ]t w-~ 0, (4.2) i l wIl~-- inf lim 1t w -- w. dw for all l w,, l Cauchy in W I~v,,l '*--~ and II w. !]v~ O, we have [fl ]1% II ]]~]e~, and /'or every other [11 H%, II Ih~]~: (4.3) 11 v I[v< II v llv, II re tlw~ II w 11~. Obviously II I]]z defined in (4.2) is a pseudonorm; we have to prove that actually it is a norm. If v eV, v=~0, and for every e>0 there is a sequence IvY} such that: tl v~,,lt~---.-o, lira IIv-v~i!,,<~, it is obviously possible to construct a sequence v,,} such that: II v,~ li w ~ o , lira ll v - v., v = O but, sinoe we suppose [V, W] compatible, this cannot happen for a v ~ 0. From the definition {4.2} of the new norms I[ il~v, II i0iw; , by taking vn----0, w,,---0, we see that 0 62 N. ARONSZAJN - E. GXGL~Am)O: Interpolatio¢~ spaces, etc. To prove that [l[ il°v, [] [t~]e~ we have furtimrmore to prove that by taking these norms [V, W] becomes a completely compatible couple. Firs~ of all the compatibility (see [2.I]) can be proved as follows: Let I[ un -- v l]°v--,-0, ]l u,~-- w lj]~..... O, (u, e V A W). For every s > 0 we can find.:n~, and according to Definition (4.2), we can find v~, w~eVA W in such a way that: I I Consequently, it (u,-- v~-- 'w~) -- v i v< 2~ and I{ (u,-- v~--w~)-- n, It ~v< 2~. Since IV, W], with norms ]l [Iv, [I liw, is supposed to be compatible, v--weVAW. We pass now to the proof of the complete compatibility. Let l u, l C [? N W be a CAucI-[Y sequence in the norms I[ H]z, [I I]~ such that ]J u,,]]~0. ~re have to prove that N u, li~~0. Without loss of generality (by replacing, if necessary, {u, l by a suitable subsequence) we may assume 1 1 According to definition (4.2) for all positive integers n, m ~e can find ~v~,,, and v,, in VOW in such a way that: 1 ilu,,~--v~[iv 1 1 lLwn, y.,.--~v...~,,ltw<~-~n for m',m"> m, J]v~}lw"~ m' 1 m2,~ • From these inequalities it follows, for m ~ n: ,~-1 1 1 ~---1 2 II (u,. -- v~ + " ~v~. ~) ]lv < -, N. ARONSZAJN - E, GAGLIARDO: Interpolation spaces, etc. 63 and for m"> m'>m>n: II (u.., - v.. + Y, ~vk..,,) -- (u~,, -- v.,,, + ~ wk. .,,,, }! w % tl v,., }1- + II v~. ;lw + II Uk -- U~:+I -- W/~,m, IW 3 I < m + ~-~ • 1 Hence by definition (4.2) II u,~ tt~ <_ 2~-,- To complete the proof of Theorem [4.I] we have to prove (4.3). Let veV, and ivy} be any sequence in VA W such that. lim tl v~ -- v,., ~'v = O, II v. ttw -~ O. By (4.1) II s,, iii~ ~ 0. Hence, by complete compatibility, II v,~ I!~-~ 0. Consequently, by (4.1), Ilviw=limPt Ilv~v,,llvr:t ~ lira llv~v,, Iv, which obviously implies the first inequality in [4.3). The second is obtained similarly. [4.[I]. COROLLARY. - Under the hypotheses of Theorem [4.I] if W is a BA:NACt]: space, then II w tl~----- ll w IIw for w~ W. The proof is immediate since now, in the second part of definition (4.21, there exists w'~ W with I[ ~v~-- w' tiw--'-0 and by compatibility of [V, W~, ~d--0 and lira tlw--w,,!Iw=ltWllw. 64 -N. ARONSZAJ.n - E. GAGLL~,RDO: Interpolation spaees~ etc. CHAPTER U. Interpolation spaces between a couple of compatible Banach spaces. § 5. The lattice of normalized Banach subspaces ofaBanach space.- Con. sider at first an abstract vector space E without topology. The class of all subsparces of E is partially ordered by inclusion. In this partial order the class of subspaces forms a lattice: for two subspaces Y and IV, the meet is given by V F~ W and the joint by V+ ]~. This lattice is complete: for any class of subspaces V~, i e I, the meet is given by (3 V~ and the joint by E l~ which is the set of all vectors ~2 v~ with v~e F~ and v~ ~ 0 for at most a finite number of indices i. From now on in this section, E wilt denote a ]~A~AOH space. [5.I] DEFINITION.- A subspace V of E is called a BT-subspaee (Banach- topologival subsp~ce) if it is provided with a BANAOH space topology such that V C E. In a BT-subspace the norm is not fixed. However, if for a subspace V there exists a topology making it into a BT-subspace this topology is unique i~). The ordering by inclusion makes the class of all BT-subspaees of E into a lattice. One checks immediately that for two BT-subspaees, V and W, if we choose two corresponding norms, then the meet and the ,joint are given by V f3 W and V-{- W with topologies defined by the norms (2.1). It is therefore a sublattice of the lattice of all subspaces. However, as we shall show in Remark [5.V], for infinite-dimensional E this lattice is not complete, not even ~-complete. Since the fact that V is a BT-subspace of E depends only on the topology of E (.but not on the choice of the norm in E) we can speak of BT-subspaces of BT-subspaees. The closed graph theorem gives immediately the following statement: [5.II]. - If V is BT-subspace of E and W is a subspace of V then W is ce BT-subspace of E if and only if it is a BT-subspace of V. ('~} This follows by ar~ immediate application of the closed graph theorem. ~. ARONSZAhN - E. GAGLIARD0: Interpolation spaces, etc. 65 [5.III] RE~AI~K.- If {Vk} is an increasing sequence of BT-subspaces, then U V~: is a BT-subspace if and only if the sequence is stationary (~). In fact, if U V~ were a BT-subspaee the Vk's would be of first category in it unless they were equal to it. [5.IV]. RE~ARK.- If {Vk} is a decreasing sequence of BT-subspaces, V ~- N Vk is a BT-subspace if the sequence t l?~} is stationary, where V k is the closure of V in Vk. The proof here is more sophisticated. With any choice of corresponding norms on ~k, V becomes, in a natural way, a FRECttET space, and by canonical identification the conjugate spaces of V k form an increasing sequence whose union is the dual of V. If V were to be a BT-subspace its ~RECgET topology would be a BA~AC~ topology and we could then apply the previous remark to the conjugate spaces. [5.V] RE~Am~.- We can show now that for an infinite-dimensional space E the lattice of BT-subspaces is not a-complete. To this effect we need only a strictly decreasing sequence of BT-subspaces, Xk, such that (~ Xk is dense in each Xk. If there were a BT-subspace X which was the greatest lower bound of the X~'s it would have to be CN Xk and, on the other hand, would have to contain all the one-dimensional subspaces generated by elements of C~Xk. It would follow that X--A X~ -- in contradiction to the preceding remark. It remains to construct a suitable sequence t Xk}. Take a sequel~ce of linearly independent normalized elements u~,, II u, llE: 1, n-- 1, 2 ..... Denote by c,, the largest positive number su~ch that c, ~ I ~kl~]l Z ~UklIE for all systems of n complex numbers ~,. k --I /¢~1 Clearly, 1 ~c~c~+~. For a sequence of positive numbers A.~ 1, con- sider the subspace X of all vectors v representable as a series Z~,u, convergent in E with z--A" l~ I ~oc. It is easy to show that for a ~n vector v there exists at most one such representation. Therefore we can put Il v llz-- ~, A,.c-~[~ [. Thus X becomes a BT-subspace of E. By choosing, for a positive integer k, An--2 '~i'~',~) we obtain the subspace Xk. Clearly, the subspaces Xk form a strictly decreasing sequence. Furthermore, since each u,,~ NXk this intersection is dense in every Xk. Thus we have con- structed the desired sequence {Xk}. (~) I.e., for some k0, V k~ Vko for k~k 0. Annalt di MaSematiea 66 ~N. ARONSZA;IN - E. GAGLIARD0: Interpoh~tion spaces, etc. [5.YI] DEFINITION. - A subspace V of E is called a Banach subspace of E if it is a B~N_~cH space continuously contained in E. It follows that a BANAC~ subspace is obtained from a BT-subspace by fixing a corresponding norm in it. A BT-subspace with two different cor- responding norms gives rise to two different B~NAc~ subspaces. For BANACH spaces we introduce the following partial ordering. [5.VII] D]~F1NI~IO~.- We write W C V if V and W are B~-Ac~ spaces, Wis a B~N~c~r subspace of V and for woW, []W!!w~]]W[!v. If W~V and W~ IT, then we wilt write W~ V; in this case V and W are equal as well as their norms. With this ordering, the class of BAN~C~ subspaces of E forms a lattice: the meet and the joint of V and W are given by VN W and V-{-W with the norms described by (2.1}. We denote this meet and joint by (5.~) vN w, v~w. [5.VIII] DEFINITION.- A BANACIt subspace V is called normalized if V~E. A system of BANAC]Z subspaces V~, iEI, is called normalized if each V~ is normalized and IT/C Vj implies V~ V i. /£ simple inspection of the norms (2.1) gives the statement: [5XX]. - If two Banach subspaces V and W form a normalized system, then the four subspaces V, W, F~J W, and V(~ W also form one. [5.X] LE~I~A. - If the system IVy}, k == 1, 2, ..., n, is normalized and V~÷~ is a BT-subspace, there exists a corresponding norm in V,+~ such that the system IV t:}, k--1, 2, ..., n, n-~ 1, is also normalized. PRoof. - In fact, we can choose to start with any corresponding norm in Vn+l making it into a BA~ACtt subspace. Consider all the indices k'1 (5.2) gives the subspace G,+l provided with a norm for which our statement is true. The last statement, by induction, leads to the~following proposition. 1~. ARONSZAJ'N - E. GAGLIARD0: Interpolation spaces, etc. 67 [5.XI].- For every countable system of BT-.subspaees we can find cor. responding norms for its subspaees which make the system normalized. We come now to the main theorem of this section. [5.XII]. T~rEORE~. - Let t Vi }, i e I, be an arbitrary system of Banavh subspaces a) there exists always the meet (~ V~; b) if there exists a Banach subspace W such that V~ ~ W for every i, then there exists the joint ~ ~. ~I P~oos.- a) We define (5.3) ve(~ V~ if yen V~ and ltvII(~-~ sup lJVllv. It is obvious that the set ~ V~ so defined is a subspaee, and that ]lvll~v. is a norm on this subspace. To prove the completeness we notice that a CAUCEY sequence {vkl in this norm is necessarily CAUC~[Y in every V~ and hence in every V~ converges to the same element v for which we then get I]v-vk]l(~v~--sup ![v--vkllvi-- sup lim Hvm~vkl]v i __~ sup supllv,~--v~llv/-- sup IIv~--v~]l~v i. Thus (~ V~ is a BANACg space and since it is clearly a BAbTACK subspace of each V~ it is also one of E. It is immediately checked that the definition (5.3) gives, in fact, the greatest lower bound of the system {V~}. b) We define the joini as follows (5.4) ~) ~ is the set of all v e E such that there exists an admissible sum Zv~ with the properties vie V~, ~A [lv~livi (5,4') II v !Iv v,. = inf ~ If v~ lJvi, the infimum being taken over all admissible 8~A~n8. ~rom the properties of an admissible sum ~,v~ it follows that at most an enumerable number of v(s differ from zero. Therefore the sum is, de 68 ~. ARONSZA3N - E. GAGLIARDO: Interpolatio~ spaces, etc. facto, an enumerable sum. Since IJv~II~ IJ v~ilw it follows that the series E v~ converges in W, and by compatibility converges in W to v. From (5.4') we then get (5.5) I] v 11~£ 11 v I1~ 5" It is clear that (~V~ is a subspaee of E and that ttvtl~ is a norm on this subspace. To prove that this norm makes UV~ a complete space, consider a CActi-ix sequence (u (~) } in it. Without toss of generality we may assume that this sequence satisfies the conditions: oo u(°)--O, ~ II u (~) ~ u(k-~) llu v~. < ~. k~Z There exist admissible sums Ev~k), k ~--1, 2, ..., such that i v~~) -" u (~) -- u (k-~) in E and ~, I1 v! ~) ttv~, < II u k -- u (~-~) lIu v + 2 -k. i i It follows that ¢2(3 (5.6) ~. :~ tl v~ k) Jlv~ < ~. tEI k~z Therefore for each i, Y~v~~) converges to some element v'~ in V~ and k ]L v'~llv~ ~ c~. Hence Ev'~ converges in W to an element u and thus also converges to u in E (since W C/~}. Therefore ue (~ i~ and we have, ff k II u -- u(~) iIu v, = it u -- :~ (u(t) -- u(~-~))Nv v. k ~ It v'~- ~ v 7) Ilv, 5=1 cx~ By (5.6) we then get Ilu--u (k)]]Uv~.--~0 for k--~. Hence ~V~ :N. ARo~szxJ~ - E. GAGLIARD0: Interpolation spaces, etc. 69 is complete. By (5.5} we have ~ W and hence UJ i~ is a BANAC]~ subspaee of E. It remains to show that it is the smallest upper bound for the system / V~!. That V~,C U)Fi for each index i' is shown by assigning to any ele- ment v ~')eV~, the admissible sum Ev~ where v~=0 for i~i' and v~--v(~') for i :i'. Suppose now that V~ V~ for every i. Then, in the, same way as we proved that W~ U) V~, we prove also that V~ U) I~. Our theorem proves that the lattice of all BA~AcH subspaces of E is boundedly complete. That it is not complete is shown by the trivial example of B~CAOl=[ subspaces E~ which are equal to E but have for norms s I[ II ~. It is quite clear that this class does "not have an upper bound. We will denote by £(E} the class of all normalized BA~AC~r subspaces of E. Our theorem gives [5.XIII] COROLLARY. - ~(E} is a complete tattice. [5.IX] RE~ARK,- We could proceed to define the joint U)V~ in the following way. Consider first the algebraic sum E Vi composed of all vectors v--Ev~ where only a finite number of v~'s ~= O. For all such sums one considers then the infimum of EII v*l!r~ and takes it as [] v ll'zv, *2V~ is then a normed space continuously contained in E. In general, this space will not be completely compatible with E. By applying Corollary [4.II] we obtain then, canonically, a new ~orm Ilvjl~v + on E Vi which makes it completely compatible with E. If we then form the completion of Y,V~ in E we obtain exactly U V~ (~). By an example we will now show that the subspace v l~ with the norm }[vtl'~v~ introduced above ,actually may not be completely compatible with E. To this effect consider an infinite dimensional BA~cAcg space E, and in this space a sequence of linearly independent elements u~ such that I] u,~ []z -- 2-'*, n---1, 2 ..... Form the one-dimensional subspaees Vi as follows: V~ is generated b$ u~ and V~ is generated by ui--u~_~ for i> 2. On ~ take the norm of E; thus V~ is a normalized BA~ACK subspace of E. The elements in EV~ are of the, form v--Evi with finite number of non-zero terms. Since the u~ are linearly independent for each such v the representation Ev~ is unique. Hence l] v !]'zy i = E 1[ viH~. It follows that for (7) In the first arrangement of the proofs of the theorems in Chapter ]IL we were using this procedure to define ~I~.. This led to the considerations and theorems of Section 4. 70 N. ARONSZAJN - E. GAGLIARDO: I'nterpol¢tior~ spaces, etc. the elements u~ = u~-~ ~ {u~-- u~_~) we get ilu,~ il'~ --][u~ lIE Jr ~llu~--u~-~llE; /:2 i=2 hence these norms increase with n. On the other hand, for n > m, i • __ i~m@:[ hence lug} is a C~go~Y sequence in the norm ]l lt'~v~ which converges in E to ~ero, whereas lira 1[ un II'z¢i ~ 0. [5.XV] REg£RK. - In proposition [5.XI] we proved that any enumerable system of BT-subspaces can be normalized. This is not true for an arbitrary system of BT-subspaces; it is not true for the system of alI BT-subspaces. In fact, if we could normalize the whole system of BT-subspaces of E they would clearly form a complete lattice -- in contradiction to Remark [5.V]. [5.XVI] DEFINITION.- Let A be a Ba~cAo~ space and ~ a positive number. We will denote by ~A the space A with the norm 11 v I[pA = a II V[[A. Thus 0A is a BANA0]t space and ~A~A, ~A, or ~D A, depending on whether ~> 1, =1, or <1. The following proposition is obvious. [5.XVII].- a) p'(~A)= (f~)A; b) f~AU,o'A --(min(,~, p')tA, ~At'~I fA--(max (~, ~'))A; c) if AcB, then ~A~2B; d) for any family of Banach subspaces of E, t V~l, i e I, (~ V~-'(~ ~V~, and if UV~ exists, then ~) V~= ~J~V~. [5.XVIII].- Let {~}, i--1, 2, ..., n, be a finite system of Banach subspaces of E such that they are mutually distinct as subspaces. There exist positiw numbers, ~1, pa, ..., p,~ st~oh that the system {~V~} is nor- malized. \ . Pnoo•. - We prove this by induction on n. If n--1, since V1 CE, the identity mapping V~--,-E is bounded with bound c>0 and we can then take ~, ~-e. Suppose that the proposition is already proved for some n~ 1, so that we have the positive numbers ?'~, ..., ~',~ making the system {~'iVi}, i----1 ..... n normalized. To prove our statement for n-~ 1, we proceed as in the proof oi Lemma [5.X] by considering the indices k', ~ k'2 <...~n ~. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. 71 for which Vk, D V.+~ and the indices k'~' < k~' ~ ... :~ n such that Vk,, C V,+~. If E is not among the V,'s, with i~n, put p'oVo~E so that 0 is among the indices k'. Denote by ck, the bound of the identity mapping V,~+~'k,V~, and by ck,, the bound of the identity mapping ~'~,,Va,,--~ T~+~. We choose then ~; = ~'~ for i ~n and i not among the indices k"; ,~.+~--max c~,; for k' i among the k", ~ ~ ~'~ (max c~,)(max c~,,). It is then immediately checked that the system !~, V~}, i ~n-~ 1 is normalized. [5.XIX] t~ES~AtlK.- The normalization procedure described in the pre- ceding proof may be considered as simpler than that described in the proof of Lemma [5.X]. However, the procedure above does not lead to a norma- lization of an arbitrary enumerable system, since in each step of the indue- tion we h,~ve to change norms established in preceding steps. § 6. The Banaeh algebra ~[V, W], intermediate and interpolation spaces for a compatible Banach couple IV, W]. - In this section IV, W] will denote a compatible couple of BA~'ACH spaces (~). With the norms (2.1), V-~- W and V(% W are BA~AC~ spaces and V, W and VA W are BA~ACH subspaces of V-l- W such that V~ W~ lz-4- W and V~ W~ VAW. For a linear continuous operator T on a BA~AC]t space A we write I TIA for the bound of T on A, i.e., the smallest constant c with ]1 Tu !IA ~ c[I u ]]A for every u e A. We will write ~'~A for the BANACtt algebra of all such operators. We recall that a mapping T of V O W into V(2 W is called a linear continuous operator on the couple IV, W] if it transforms linearly and continuously V into V and W into W. Such a mapping has a canonical extension to a linear mapping of V+ W into V+ W; also it linearly transforms VC~ W into VN W. [6.I] - If T is a linear continuous operator on [~/, W], then (6.1) [TIv+w~max (IT[v, [ TIw), I Trvnw PROOF. - For ue V-4- W, we have, by (2.1}, II Tullv+w~ inf (ltTvlivd- +llTwltw)£ inf (tT]vliV]]vd- [ T twllWllw) which proves the first part of (6.1). If ue VNW, then [] Tullyaw-- max (H Tullv, l] Tullw) S~ ~_ max (I TIy I1 u [iv, ] T Iw [I u []w), which proves the second part. (s) We remind the reader that for BANACH spaces, ,,compatibles) is equivalent to completely compatible ,,. 72 N. ARONSZA;IN - E. GAGLMaDO: Interpoh~tion spaces, etc. [6.II]. DEFInition. - We denote by ~[V, W] the class of all linear continuous operators on [V, W]. For Te~[V, W] we put I Tl~;iv, w]-- max( tTlv, ITIw). In the present chapter the couple [I~ W] will be fixed and we will write, briefly, ~ and 1 T I ~ for ~;[ V, W] and I T l~;[v, w] respectively. The proofs of the next two propositions are immediate and we omit them [6.III] - ~ ~[V, W] is a Banach algebra with the natural operations aS~--~T, ST (~, ~ scalars, and S, T i~ ~). and with the norm I Tt~. (9). ~; is a sub-algebra of the Banaeh algebra of all bounded linear operators on V 27 W in which it is continuously contained. [6.IV]. - Consider the direct sum of the Banach algebras ~v and ~w, ~;v-~Cgw, with the norm I] iT~, T2} ll--max (I T~Iv, I Tz[w). For each Te¢~, define T~ and T2 as restrictions of T to V and W. The mapping so established, of ~ into ~z-~ ¢gw is an isomelric isomorphism of Banach algebras. The range of this mapping is the set of all co~ples t1'1, T2} e~v ~-~wi satis- fying the condition (6.2) For ue VQW, T,u= T~uE VQW. [6.V] D~FINITIO~.- An intermediate space between V and W is a BAtCAe~ subspace of V 27 W containing VQ~. A is a normalized inter. n~ediate space between V and W if V27 W~ADVQ W. [6.VI] - a) Each intermediate space A can be provided with an equi- valent norm which normalizes it; for instance, one can choose the norm of [AC~(V + W)]U(~Q W~. b) All normalized intermediate spaces between V and W form a complete lattice under the ordering C. The proofs follow immediately from Lemma [5.X] and Theorem [5.XII] respectively. ~Notiee that if VV~ W~ 10), and we put only the restriction V+ W~DA on the intermediate spaces A, the resulting lattice would not be complete (not even ~-complete). [6.VII] DEfinition. - An intermediate space A between V and W is called an interpolation space (between V and W, or in [V, W]) if T(A)cA (~) In particular we have f ST I ~; ~ I S I ~ ! T I ~7" N. ARONSZAJN - E. GAGLIARD0: Interpolation spaces, etc. 73 for every T e~. An interpolation space A is called normalized if it is a normalized intermediate space and satisfies the condition (6.3~ t TIA [6.VIII] REMAnK.- We could try to introduce a more general notion of an interpolation space A by requiring, only, that it be a BAl~ACI~ sub- space of V+ fi/ with the property T(A)CA for every TeE. However, under these conditions, except where A--(0), A must be an intermediate space. In fact, if 0 @ a eA C V+ l~; there exists a continuous linear functional f(~c) on V+ W such that f(a) = 1. If u is an arbitrary element of g(~ W, the iinear operator Tx=f(x)u clearly belongs to E. Hence u = TaeA. By Proposition [6.1] we have [6.IX] - V + W and V (~ W are normalized interpolation spaces between V and W. [6.X] LE~t~[A. - Let E be a Banach Space,and A, B two Banach sub- spaces of E. If Te~E and T(A}CB then T transforms A contiuuously into B. PROOF. - By the closed graph theorem it is enough to show that the restriction TA of T to A is a closed mapping of A into B. Suppose that a~ ~a in A and Ta,--~ b fin B. Then, since A CE and BCE, it follows c that a,-+a in E and Ta,.--*b in E. Therefore b--Ta, which finishes the proof. [6.XI] THEORE~L - Let A be an interpolation space between V and W. There exists a constant c >O such that t T~]A <~C I TI~; for every TeE, where TA is the restriction of T to A. PROOF. - Since, by Proposition [6I], T~v+~, the preceding lemma gives T.4~ ~A. Thus the theorem states that the linear mapping T---~TA of into EA is continuous. Again, by the c]ose~d graph theorem, it is enough to show that if T(k)-~ T in ~ and T(Ak)--.-S in ~, then S--TA. To this effect consider any element aeA. Then clearly T( k) a--~ Ta in V+W and,T [6.XII] - If A and B are interpolation spaces between V and ~¢¢ so are also A~B and A(~B. Thus the class of all interpolation spaces between V and W is a lattice; it is actually a sublattice of the lattice of all inter. mediate spaces. Annali di Matemativa 10 74 N. ARONSZA~ - E. GAGLIARDO: Interpolatio~ spc~ces, etc. The proof is immediate. [6.XIII] TI-IEOI~E~L - All normalized interpolation spaces between V and W form a complete lattice which is a sublattice of the lattice of all norma- lized intermediate spaces. P]~ooF. - 0nly the first part of the theorem requires a proof. Let tA~!, i~I, be an arbitrary family of normalized interpolation spaces. We have to show that ~ Ai and UA~ are normalized interpolation spaces. Since they are normalized intermediate spaces, we need only prove that T(f~Ai)C(~ As and T(UJA~)C UJA~, and that I T[(~A.~[ T[ v and [ TIUA <~ [ TIV. Suppose aeriAl. Then aeAi, ieI, and [[all(~,~i:sup []a[[A{ a) For every positive ~, ~A satisfies (6.3). b) If B is a normalized interpolation space, then AUJB and Af~B also satisfy (6.3). c) If B and C are normalized interpolation spaces then (A (~ B)U)C is a normalized interpolation space. PRoof.- a) is obvious, b) follows from Proposition [6.I] if we re- place [V, W] by the couple [A, B] and notice that for Tecg[V, W] we have I T[A~[ 7'[V and ] TtB~I TIV. To prove c), we remark that the BAI~AC~t subspace D-----(A(~B)~C satisfies (6.3) by virtue of part b). On the other hand, since V+W~B~ VN W and V+W~C~ VNW, then also D satisfies these inequalities. [6.XV] - Let A be an interpolation space. Define for a e A, (6.4) II all'A---sllp [ITalIA, the supremum taken over all T e ~ ~vith l TI v <_~ l. 11 a]t'a is a norm equivalent to the original [I aliA. The space A, provided with the norm (6.4) satisfies condition (6.3). (i0) A may or may not be a normalized intermediate space. N. ARONSZAJN - E. GAGLIAI~DO: Interpolation spaces, etc. 75 PROOF.- For a~A, denote by a the linear transformation of ~; into A defined by aT--Ta. Clearly II a Il'A --" [ a[a,~; (x~). It follows that ilaH'a is, in any case, a pseudo-norm. On the other hand, by applying a to the identity mapping we get [alA,~ ~ [I alia and by using Theorem [6.XI] and (6.4) we get [I,a 1t'.4 ~c ][ a[[A. Hence (6.4) gives a norm equivalent to I[ a Ila. Finally, to prove condition (6.3) it is enough to consider it for opera- tors SeCg with ISI~=I" We have then I! 8a ll' = I1TSa liA < sup it T'a lIA = I1 a II'A. I TiV~_t IT'iV_~ [6.XVI] COROLLARY.- Let A be an interpolation space between V and W. We write A' for the space A provided with the norm {6.4). Then the space [A'(~ IV-}- W}]UI V(~ W) is e~ual to A and h~s a norm making it into a normalized interpolation space. The proof follows immediately from [6.XV], [6.IX], and [6.~IV, c)]. [6.XVII] THeOREm. - a) If A and B are interpolation spaces between V and W and C is an interpolation space between A ancl B, then C is an interpolation space between V and W. b) If A and B axe normalized inter- polation spaces between V and W and C is a normalized interpolation space between A and B, then C is a normalized interpolation space between V and W. PROOF.- It is clear that [A, B] is a compatible couple of BA~,~CH spaces and that for TEcC[V, W] the restriction of T to A-[-B belongs to CC[A, B]. From this remark a) follows immedialely. Under the conditions of b) it is clear that C is a normalized intermediate space between V and W. Furthermore, I TIc~ max (I T]A, I TIB) ~ ] Ti,~;'. [6.XVIII] RE~ARK.- If A, B, C are interpolation spaces between V and W and C is an intermediate space between A and B, it does not follow, in general, that C is an interpolation space between A and B. § 7. The ease when VNW is not dense in V+W. - For a subset C of a space A the closure in A will be denoted in general by -A.C For brevity, the closure in the space V-{-W will be denoted C. As in the preceding section IV, W] will be a [ixed compatible couple of BA~ACH spaces. (ii) ]~or two ]SANAOH spaces X, Y, and a linear mapping S of X into Y we denote the bound of S by 1SIY, X. 76 :N. ARONSZMN - E. GAGL:[ARDO: Interpolation spaces, etc. [7.I] LE~MA. - Put (7.1) V~ = ( v n w) v, W~ -- (vn W) W. Then the follotving relations hold. (7.2) F= v+ w:, = w + v:, (7.3) vTn = (Vn W)= PaooF. - In the canonical mapping of V+W onto V+ W (see (1.4)) the inverse image of V is clearly V~- (VN W). The closure of the last set in V~W is therefore the inverse image of V. Hence, we get for this inverse image the subspace V+ W~ which leads to the first formula in (7.2)-; the second formula is obtained similarly. Since the topologies in V and W are finer than in V-~ W, we have VNWD(VN W) DV~+W~. On the other hand, by (7.2), VNW -- (V+ W~) n (v~-{- w) -- v~+ w~, which gives (7.3). The notation (7.1) will be maintained through this section. V~ and W~ are closed subspaces of V and W respectively and will be provided with the norms of V and W. V, W, and VNW~- V~+ W~ are closed subspaces of V+ W and will be provided with the norm of V+ W. [7.II] - 111, W~ V, W, and V~-W~ are normalized interpolation spaces between V and W. The proof is obvious. [7.III] LEPTA.- Let A be an interpolation space behveen V and W. a) If A C V then A D W ; b) if A C W then A D V. PROOF.- It is enough, obviously, to prove a). If A~: ~ there exist a and % such that: 0~aeA, a~ V, ~ is a bounded linear functional on V--}- W vanishing on l/, and %0(a)-- 1. Let w be an arbitrary fixed element of W. The linear mapping Tu= ~(u)w beloflgs clearly to ~C. Hence w = 7'ae A which finishes the proof. [7.IV] THEOREm.- If A is an interpolation space bet~veen V and W, ~*. ARONSZAJ'N - E. GAGLIARDO: Interpolatio~ spaees~ etc. 77 it must satisfy one of the four conditions: 1o) A-~ V+W, 2 o) FCAC~ ?, 3 o) WcAciV, 40) VNWCAC V~+ W,. PROOF. There are four mutually exclusive possibilities for A. 1o) A@~" and Ac~ W; 20) A ~ l~ and AC :~Y; 3 °) A@ V and ACI~; 40) A C V and A C W. By the preceding lemma these four possible situa- tions lead to the four corresponding cases of our theorem. An immediate corollary of this theorem is the following. [7.V] Coao~L~t~Y.- The only interpolation spaces between V and W which are closed subspaces of V+ W are V~+ WI--(VN W), V+W~-~ V, W+ V~ -- ~I-z, and V -[- W. [7.VI] RE~ARK.- a) Suppose that V~ V and W~ ~ W, i.e., that VC1W is neither dense in V nor in W. The four cases of Theorem [7.ZV] are mutually exclusive and represent disjoint classes of interpolation spaces between V and W, exhausting all of them. bt Suppose now V~--V and W~ ~ W (or, symmetrically. V~ V. ~t~\ : W), i.e:, that v~n w is dense in V but not in W. The classes of interpolation spaces represented by 3 o) and 4 0) are disjoint and contain those represented by 1°) and 2 0) respec. tively, c) Suppose finally that V~ : V and W~'= W, i.e., VN W is dense in both V and W (and then also in V~W). The classes of interpolation spaces corresponding to 1°), 2°}, and 3 ~} are contained in the one corre- sponding to 4o). [7.VII] TgEORE~.- a) If AI is an interpolation space between V~ and W~, then V-b A~, W~AI and AI are il~terpolation spaces between V and W corresponding to cases 2o), 3o), and 4 °) of Theorem [7.1¥] respectively. b) Assume that the following property holds. (7.4) Each mapping from ~vl and ~dw~ has an extension to a mapping in ~dv or ~w respectively. Then for any interpolation space A between V and W, AI-~A (~ (VI+ W1) is an interpolation space between V1 and W1 and if A belongs to the case 2 °) or 3 °} or 40) of Theorem [7.IV], we have A:V-I-A1 or A-~ W-i-A~ or A ----At, respectively. 78 ~N. ARONSZAJI'¢ - E. GAGLIAI~DO: Interpolation sp,aces~ etc. PRoof.- For part a) we have only to refer to Theorem [6.XVII], Proposition [6.X[I] and formula ~7.2). Assuming the hypotheses of part b), we notice that for every Te ~:[V:, W:] there exist extensions T~CCv and T2+CCw of the restrictions Tv, and Two. Since V~N W1-----VN W and 7'v~ and Tw~ satisfy property (6.2) relative to V: and W~ (see [6.IV]), this property also holds for 7'1 and T2 relative to V and W. It follows by [6.IV] that T~ and T2 are restrictions of an operator 7"erC[V, IV]. A~ being an interpolation space between V and W, it follows that T(Ax)--T'(A~)C A~. The remaining statements of part b) result immediately from formula (7.2). [7.VIII] REMARK. - We d~o not know if, property (7.4) is necessary for the validity of part b) of the last theorem. The property < (7.4't There exist continuous linear projections of V onto V~ and of W onto W~. This proper~y holds, for instance, when V and W are I-~I:LBERT spaces. Another example where this property holds is when V- L*(R~), W--the class of all totally finite signed BOREL measures in R"; we have here vn W = L~(R'~)NLI(R% V~ -- V and W1 -- L~(R ~) [7.IX] REMARK. - The interest of Theorem [7.VII] lies in the fact that when it is valid it allows us to reduce the study of interpolation spaces to the case when VN W is dense in V and W (hence also in V+ W). Almost all interpolation methods in the literature produce classes of interpolation spaces depending on one or several continuous parameters. These interpo- lation spaces are in .the case 4 o) of Theorem [7.IV], and for most of them it can be proved that they are interpolation spaces between ]71 and W1. [7.X] RE:~[ilC,K.- We finish this section by a simple illustrative example. Suppose that VNW is a closed subspace of V as well as of W. It follows then by (7.2) that V, W and vn W are closed subspaces of V+ W. To each of the cases of Theorem L7.IV] there corresponds only one interpolation space: V+W, V, W and VN IV, respectively, Consequently, between V+W and VNW there are only two interpolation spaces: V+W and VNW. Hence, in this case, V and W are not interpolation spaces between V+W and VNW. This is a special case of a general result {see Theorem [lO.XVI]. N. ARONSZaaN - E. GAGLIARDO: Interpolation spares, etc. 79 § 8. Conjugate couples. - As in the preceding section IV, W] is a fixed compatible BA~AC:~ couple. We shall investigate the conjugate spaces V*, W*, especially in cases when they form canonically a compatible couple. We recapitulate first the way in which the BANACI~ spaces VNW and V-~-W were formed in Sections 1 and 2. We form Vq-W and its closed subspace, Z-- [{v, w!: w----v]. Putting the norm II {v, w} fl -- max { I] v]Iv, ]] w]'w) on V4- W we get a canonical isometric isomorphism of -VQW onto Z: u~Iu, --ul. Choosing the norm L[~v, wlH=!lv~fvq-llwllw we obtain a canonical isometric isomorphism of Vq--W onto (V4-W)/Z: u=v ÷ ~v~{v, wt + Z. We will use the following well known theorems. 1o) The conjugate space of a direct sum is the direct sum of conjugate spaces. If the chosen norm is max (I] v ltv, il w lIw), then the conjugate norm is II v* t~v*l~-k I] w* l!w*, and if the chosen norm is !l v l[v q-llwlIw, the conjugate norm is max(ll v*tlv, , Hw*!lw.). 20) If A is a closed subspace of a BA~rAO~t space E then A*~E*/AI, and (E/A)*~AI, where Al is the the subspace of functionals from E* vanishing on A. It will be convenient for us to define the pairing between V4-W and V*4-W* by the scalar product, (8.1) <{v, w}, {v*, w'i>-- this deviation from the usual definition by sum obviously does not change any of the previous statements. One checks then immediately that (8.2) Zl--[{v*, w*}: The preceding remarks lead immediately to: [8.1] - a) The conjugate space of V+W is canonically isometrically isomorphic with ZI , with norm ]l {v*, w*} I[--max (I]v*][v*, IIw*[]w*), the scalar product being the one induced by {8.1). b) The conjugate space ot YN W is canonically isometrically isomor. phic to (v*q-w*}/zl with the norm induced by the norm llv*IIv*q- q-[] w* J[w* on v* q-w* and scalar product induced by (8.1). The comparison of the definition of Zl by (8.2} with the definition of 80 ~-~. AI¢0NSZAJN - E. GAGLIARD0: Interpolo~tio~ spaces, etc. Z leads in a natural way to the following identification of elements in V* and W*: the linear funetionals v*eV* and w*eW* will be considered equal if they have the same restriction to VNW. However, this identifi- cation is only acceptable if it does not identify different elements of V* (or W*) with the same element of W* (or V*). 0ne sees immediately that this condition is satisfied if and only if VNW is dense in both V and W. We are thus led to the following definition. [8.11] DEFI:NITION.- If (8.3} V(5 W is dense in both V and W, V* and W* form a linear couple IV*, W*]- the conjugate couple- with the identification mapping defined as follows" (8.4) v*~-w* if and only if When we speak about the conjugate couple or use the symbol [V*, W*] it will be assumed that condition (8.3} is satisfied. From Proposition [8.I], by virtue of the last definition and formula (8.2), we obtain [8.III] TKEORE~.- If the conjugate couple IV*, W*] exists, then (V + W)*~ V*N W*, (VN W)*~ V* + W*. The corresponding scalar products are those induced by the scalar product (8.t} on the direct sums V-~ W and V* ~- W*~ i.e.~ ~v+~v, u*)v+w-~ ~v, u*".~.~ + < w, U*>w (8.5) < u, v* + w* ~ vn w -- < u, v* -~ v + < u, w* > w. An immediate consequence of this theorem is: [8.IV] COROI~L~R¥. - 1[ IV*, W*] exists then : a) If W C V {i.e., W is a Banach subspace of Y( then Y* C W*. b) If WCV, the~ V*~ W*. et V~V~W* is dense in V*, W*, and Y* + W* in their respective weak*-lopologies (12). d) If V and W are reflexive, then V*NW* is dense in V*, W* and 17. + W* {in their normed topologies}, and IV**, W**] exists and is equal to [V, W] (in the usual identifications). (tz) But not necessarily in their normed topologies. N. ARO~'SZAJ~ - E. GAGLIARDO: I nterpolatio~v spaees~ etc. 81 [8.V] REI~ARK.- If V and W are not both reflexive, the density condition may or may not be preserved in the conjugate couple. Simple illustra- tive examples arc the following: 1°) V--L~(O, ~) and W-'L~(0,1); here the density is preserved. 2 e) V:L2(0,1)and W-----L~(0~I); here the density is not preserved. As long as the density is preserved we may form the conse- cutive conjugate couples [V*, W*], [V**, W**], etc. We do not know of an example of a non-reflexive couple IV, W] for which the fourth conjugate exists. That the third conjugate may exist is seen by taking 1 W--[t~,!: nl~,~l~0] with Ilt~t[Iw---max ln~,~[. We have here WCV. Consider an intermediate space A between V and W. A continuous linear functional on A is, afortiori, a continuous linear functional on VOW. This gives a natural identification of A* with a subspace of (VOW)* on condition that two distinct elements in A* are not identified with the same element. It is obvious that this restriction is equivalent to the condition (8.6) V• W is dense in A. If this condition is satisfied and [V*, W*] exists, the identification is valid and A* becomes a subspace of V*~ W* containing V*C~ W*. The following proposition is then immediate. [8.VI] - If A is an intermediate space between V and W satisfying (8.6), and if [V*, W*] exists, then a) A* is an intermediate space between V* and W*. b) If A is normalized so is A*. Consider now an operator TecC~_~[V, W]. Consider further the re- strictions of T, TvE~v, Twe~w and their adjoints T~.erCv,, T~v~'~w.. For ueVOW and u*e~7*OW * we have by (8.4), ~ u, T~zu* ~v-- ~ Tvu, u* ~v--- ~ Twu, u* ~w-- ~ u, T~vu*~w. Hence by Definition [8.II] and Proposition [6.IV], T~. and T~v are restrictions of a unique operator T*~75[V ~, W*]. T* will be called the ~-adjoint of T and the class of all such adjoint operators will be denoted by ~C*. AnnaU di Matematica 11 82 N. ARONSZAJN - E. GACLIARDO: I~terpolatio~ spaees~ etc. [8.VII]. TKEORE~ - a) The class ~* of all ~-adjoint operators is a Banach algebra; it is a subalgebra (with the same norm) of ~[V ~, W*]. b) The mapping T--* T ~ is an isometric anti-isomorphism (~) between the Banach algeb~:as ~ and ~. c) The operator T ~ on V ~ ~- W ~ is the adjoint of the restriction Tr'NW ; the restriction T~,nw. is the adjoint of the operator T on V ~ W. d) In order that an operator L e ~ [ V ~, W ~] belong to ~ it is necessary that it be weak* continuous on each of the four spaces V*, W*, V* (3 W* and V* -[- W* ; it is sufficient that it be weak* continuous on V* ~ W*. e) ~* =~[V*, W*] if and only if V (3 W is reflexive. f) If V A W is reflexive, then [V*, W*] satisfies (8.3)and [V**, W**] exists. PROOF. - Parts a) and b) are immediately checked by the use of elemen- tary properties of adjoint operators (such as I TIA-----! T*]~., ete). To prove c) take v* -Jr- w* e V* ~ W*, ue V N W, u* G V* ('1 W*, v -1- w ~ V -k W and use (8.5). This gives < Tu, v*-bw* > v Nw~- ---- < u, T*(v* Jr- w*) >vnw, < T(v + w), u* >r'+w= < Tv, u* >v 4- < Tw, u* >w-- < v A- w, T'u* >v+w" The first part of d) follows immediately from part c) since an operator on a conjugate space is an adjoint if and only if it is weak* continuous. For the second part assume that L is weak* continuous in V* "4- W*. It is then the adjoint of some operator He~r, nw. Hence, for any u~V A W and v* ~ V* we have < ttu, v* >rnw---- < u, Lv* >raw, and, since Lv* ~ V*, (13) Since, in this paper, we use bilinear scalar products (and not hermitian as in HILBERT spaces), the mapping T ~ T* is linear (and not anti-linear). The prefix • anti of ~ anti-isomorphism between BANACK algebras ~ refers to the property (ST)*~ T'S*. ~. ARONSZAJN - E. GAGLIARD0: Interpolatio~ spaces, etc. 83 To prove e) assume first that V (3 W is reflexive. Then V*-}= W* (V f~ W)* is reflexive too and every bouncied operator in V* ~- W* is weak* continuous. I-Ience, by part d), our assertion follows. Assume next that VCI W is not reflexive. The same is true then for V* "4-W* and there exists a bounded linear functional f(~v) on V* "4- VP* which is not weak* continuous. Taking any u* ~ V* (3 W*, u* ~ O, and putting Lx ~ f(a~)u*, we get a linear operator in ~[V*, W*] which is not, in ~* by part d). Finally, we obtain f) by noting that since V* n w* is weak* dense in V* -{- W* (see [8.ZV, v)]) if V* -{- W* is reflexive, V* n W* must be dense in it in the weak topology and therefore also in the normed topology. [8.VIII] RE~IARK - We give here a simple example of IV, W] such that IV*, W*] exists, V N W is reflexive and each of the spaces ]?, W, V~ W is non-reflexive. Take two disjoint bounded intervals /1 and /2 on the real line and consider functions f defined on I~ U [2. Put V-----If: feL ~ on /1 and f e L ~ on h], W----If: f ~ L ~" on 1~ and f ~ L ~ on I~]. Then V n W -~ L2(h O I2) and V~ W--L~(I~UI2). If we now replace V by the space V-~[f:feLP on /1, 1 [8.IX] RE~A~: = As a complement to Remark [8.V] we notice that if one of the spaces V, W, V(3W is reflexive, whereas V-~W is non-reflexive, there can exist at most two conjugate couples (otherwise V~ W would be dense in (V ~ W)** in the normed~topology). It follows that if V ~ W is reflexive and one of V, W, V N W is non-reflexive (hence V ('1 W non- reflexive) there can exist at most three conjugate couples; the maximal number may be attained as in the last example of Remark [8.V]. [8.X] - Suppose that [V*, W*] exists, A is an intermediate space between V and W satisfying (8.6) and T e ~ [V, W] satisfies T(A)C A. Then the restri- ction TA belongs to ~A and the restriction T'A, is the adjoint of TA. PROOF.- The first assertion follows from Lemma [6.X]. To prove the second assertion, we write for ueV (~ W and a* cA*, < TAU, a* >A Z < Tu, a* >vnw---- (~4) To see this we apply the theorem : If E is a BANAC~ space and A a closed subspace of E, then E is reflexive if and only if A and E/A are reflexive. We then put E-----V-~ W and A~--Z~--- [Iv, w}:~v~--v]. 84 N. ARONSZA:IN - E. GAGLIARDO: I~vterpol,ation spaves~ etc. This expression is a bounded linear functional in u with the norm [lu[lA (since TA~A), hence, by (8.6) is uniquely extendable to A, i.e. T'a* cA* and < T,~u, a* >A--= < U, T'a* >~. Again by (8.6) it follows then < T.~a, a* >A---- < a, T'a* >A [or all a EA which finishes the proof. [8.XI] TIIEOREI~I - If [V*, W*] exist8, ]7 N W i8 reflexive and A is an interpolation space between ]7 and W satisfyi+~g (8.6) then A* is an interpolation space belweel, V* and W*. If A is normalized, then so is A% The proof is clear by virtue of [8.X], [8.VI] and Theorem [8.VIII. For use in the next theorem we introduce the following notations. [8.XlI] DE~I~ITIO~¢ -~[V, W] and ~[V, W] will denote the classes of intermediate or interpolation spaces (between V and IF) respectively which satisfy (8.6). The subscripts ~¢n >>, ~ r >>, or << nr >> will mean that the classes are restricted to << normalized >>, < [8.XIII] DUALITY ~HEOm~I - Let V and W be reflexive and V A W dense in V q- W. Then: a) the same is true for the couple IV*, W*] and IV**, W**]: [V, W]. b) The mapping A ~ A* is one-one and order reversi~g on each of the classes ~, IV, W], 5,~[V, W], ~, IV, W], ~,,.[V, W] and transforms them onto the corresponding classes formed for [V* , W*] for each of these classes the inverse mapping is given by A* --* A* * ~ A. § 9. Some properties of the Banach algebras. ~[V, W] and ~;a[V, W]. Consider an operator T e ~-= ~[V, W]. If A is an interpolation space then the restriction, TAe ¢GA. We may then consider the inverse (if it exists) of T in ~ and TA in ~A. [9.I] TFLEOREI~I - Let T e ¢G, then: a) If T -~ exists, then Tdl exists for every interpolation space A and is the restriction of T-1 to A. b) If T-~ 1, T~) and T-~ W (or T~v~) exists, then T -1 exists. c) If T-~ W and T-~r¢ exist, then T -~ exists. (l~) Which in general are not complete, N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. 85 Pnoot~.- a) If T -~ exists, then (T-~)ae ¢gd and clearly equals (T~)-~. b) If T~ ~ and Tw~ exist, we have to check if they form a transforma- tion in ~. By [6.IV], it is enough to check that if ueVQ W, then T-~u----- Tw*U~ V~ W. If, in addition, we assume that T'~ W exists, then u =TvnwU, for some u~ ~ V VI W and hence T~u --~ T-~Tv~ ~u~ ~- T~,~T~u~ --- u~ and similarly, Tw~U----u~. We check then immediately that the combined transformation, T-~(V-3cW } : T~V 2V T-~lw is, in fact, the inverse of T in ~. If, instead of the existence of T~NW,-~ we assume that of Tv+-~~, then for any ueVN W, we have Tv+w(T-~u)--Tv(T-~u)----u and, similarly, Tv+~7(T-~~u) = u; hence T-~u -- T'v~wu -- Tw~U. c) If T~ and Tv-~w exist, then clearly, T~ is the restriction of T~v to V (5 W. It is enough to show that T~w(V) C V and T'~w(W)C W. In fact, for v e V let T~v -- v~ + w~, v~ e V, w~ ~ W. Then v = Tvv~ -~- Tww~. Hence, Twwz ~ V N W. It follows that T-~hwTww~ = T-~wTz+~w ~ .~ w~ and thus w~ e V C~ W and T~wVe V. The proof of T-{~w(W)C W is similar. [9.1I) RE~[ARK- In general the existence of T -~ does not follow from the existence of T~ ~ and Two. For example, consider the L~-space on the circumference I~l "- 1. As V and W take the closed subspace of L~[]~I-- 1] which vanish for Re~ < 0, or Re~ > 0 respectively. We introduce an iden. tification mapping x between V and W as follows: w--xv if there exists in I~1 < l an analytic function f(~)eI12 such that f(~)= v(~)for t~t = 1, Re~>O and f(~)=--w(~) for I~1--1, Re~ [9.III] DEFIniTION - For T e ~ we will denote by ~(T) the spectrum of T in the BA~AC~ algebra ~. If A is an interpolation space between V and W, we will denote by ~A(T) the spectrum of TA in the BA~Ac~ algebra ~;A- [9.IV] T~EORE~- Let T e ¢g. Then: a) for any interpolation space A, •A(T) C a(T). b) v(T) ---- ~v(T) t) ~w (T) U ~vn w (T) -- +v (T) tJ ~w (T) L9 ~v+w(T) -- a vnw(T) tJ ~v+w(T). v) C being the complex plane, the following statement is 86 N, ~RONSZAJN - E, GAGLIARDO: l nterpo~a,tion spaces, etc. true : Each component of C-- ~(T) is equal to a component of (C -- ~v(T)) N (C- ~w(~)). PnOOF.- a) and b) fotlow immediately from Theorem [9.I] applied to T-- ~I (I being the identity). To prove c) we denote by R(T; ~) the resolvent operator of T in ~, and by RA(T; ~) the resolvent operator of TA in ~A (for an interpolation space A). These are analytic operator-valued functions of ~, regular in the open sets C--<;(T) and C- ~A(T) respectively. Since, by a) of our theorem, C--~(T)C(C--~v(T))A(C--~w(T)), what we have to prove is that if G is a component of C-- <;(T) and G' is the component of (C-- ¢;v(T)) n (C-- aw(T)) containing G, then G-- G'. In fact, if this were not true, there would exist a ~'e G', such that ~' belongs to the boundary ~G. There would then exist a sequence i~,, !C G with ~,, -~ ~'. For each ~,,, Bv(T; ~,) and Rw(T; ~.) combine together to form R(T; ~,~). Since ~'e G', it belongs to the resolve~nt set of Tv as well as T~, and the operators By(T; ~;,) and Rw(T; ~,,)form CAUCI~Y sequences in the respective aigebras '~v and ~w- Hence the ope- rators R(T; ~,) form a CAucHY sequence in the algebra ~. This, however, is impossible since by a well known property of resolvent operators, if ~' is 1 in ~(T~, then IR(T; ~,,)[V~_~ i~,--~'l " [9.V] COI~OLL~RY- If T e ¢G, A is an interpolation space between V and W, ue(V -5 W)--A and Tu:=ku-Sa with aeA, then l II <-~tT[~. PnooF. - If ), were > I T ]~ then (T -- kl)- 1 would exist ; hence (T -- ),I)~ 1 would exist too, and would be the restriction to A of (T--XI) -1. It would follow that u--(T- kI) la = (T- klJA'aeA against our assumption. [9.VII DEFINITION - Let A be an interpolation space between V and W, We denote by ~A_-- ~A[V ' W] the closure in ~ of the set of all T e '~ with T(V~- W) CA. As norm in ~A we take the norm of ~. [9.VII] - If for an interpolation space A, T e ~A, u~(V+ W)- A and Tu=ku ~-a with ae A, then k--O. PROOF - By definition, there exist T, ~ ~ with T,(V -5 W) C A and !T -- T, I~7 ~ 0. It follows that (T -- T,)u -- ku -5 (a -- T,u). Since a - T,u e A, we get,. by Corollary [9.V], i kl ~ IT-- T. l~. ~. ARONSZAJN - ~E. GAGLIARDO; Interpola, tion spaces, etc. 87 [9.VI]I] THEOREM - Let A be an interpolation space between V and W, strictly smaller than V ~ W. There ¢~A is a prolJer closed ideal in the Banach algebra ~. PROOF.- From the definition it follows that c~,4 is a closed linear subspace of ~ and that the set of all T e ~ with T(V-[-W)CA form a dense subspace ~A of ~A. If S~ and $2 are any two operators in ~ then obviously T~ ~A implies S~TS~ ~A hence, by continuity, T~ ~A implies S:TS2 e cG n. Therefore ~A is a closed ideal. To show that this ideal is proper we first notice that for any q~(V-k W)* and u~VAWCA, the operator Tx ~ ~(x~)u belongs to ~A, and hence ~A:~:(0). On the other hand I~ ~n since for u ~ (V "t- W) -- A, Iu -- u q- 0 -- in contradiction to [9.VII], § 10. Relative completion. - The notion of relative completion was intro- duced recently by E. GAC~LIARDO in his study of special interpolation methods [4]. [10.I] DEI~II~ITION - Let /~ be a B~NAC~ space and A a BA~AC~ subspace of E. The completion of A rel. E, denoted by A(~), is the set of all elements ~eeE which lie in the closure in E of a bounded sphere in A, i.e., 2~(E)- t2 SA (R)E. For x ~ ~t(E) we put R>o (10.1) li x ItX(E) : inf R for all R with x e SA(R)E . [10.II] REMARK - ,~(E) will not be confused with the completion of A in E (denoted in § 3 by ,~E) which is obviously equal A. [10.III] - A being a Bana('h subspace of the Banach space E, we have a) ~(E~ is a subspace of E and with norm (10.l) is a Banach subspace of E. M b) A (~) ~ A. c) ~(E) ~ (completion of A tel. AE). d) The space ~t(E) and its topology depend only on the topologies of A and 1~ and not on the choice of their norms. e) The norm of : (E) depends only on the norm in A and the topology of E (and not on the choice of norm in E). 88 N. ARONSZA;~N - E. GAGL~ARDO: Interpolation spa,ces, etc. f) If B is another Banach subspace of E and A C B then A(~) C A(~) and .~(~) C I~(~) ; if A ~ B, then ~4(~) ~ [~(~). g) The space 71(~) and its norm are not changed if in their definition the strong closure S~(R) ~ is replaced by the closure of SA(R) in the weak topology of E~ i.e., by SA(R)~'~. h) (Completion of A¢~) tel. E)~ A(E). PRoo~~. - a) That .~(E) is a subspace and ]]xIIx~E ) is homogeneous and satisfies ~I~KOwSKI'S inequality follows immediately from the observation that if x~SA(R) E and yeS~(R~) E, then ~x-~ ~yeSA(R]~t-~ RII~[) E. Further- more, since there exists a constant c such that !lu!lE~ cllul!.4 for ueA, it follows that II~c!tE~-- cll~cIiX(E)- Hence, (10.1) defines a norm and A~) C E. It remains to show that _~ b) Obvious, since aeS~(Hal~) ~ C S~(tlaIIA) ~ for any aeA. The properties c), d), e), and f) are also obvious consequences of the definition of .4(~). g) is proved by noting that SA(R) being a convex set, its weak and strong closures are the same. h) Denote by A1 the completion of ~(E) rel. E. Since by b), AI~-~ [10.IV] - /t(E) -- E if and only if A -~ E. PROOF.- It is enough to show that A ~ E implies z] ~E) =~ E. In this case the identity mapping I: A ~ E is a continuous isomorphism o[ A into E (and not onto). A well known property of such mappings is that a sphere ~T. ARONSZAJN - E. GAGLIAItDO: Interpolation spaces, etc. 89 GO in A is transformed into a nowhere dense set in E. Since ~(E)-- U SA(n) E our conclusion follows. We introduce the following notation to be used in the next theorem. If X is a subspace of a BANACI-I space B, the polar set of X, i.e., the set of all elements u*eB* such that [10V] TI~IEOI~E~[- Let A be a dense Banavh subspace of the Banach space E. Then with the usual canonical identifications we have ~I(E> = E (5 (A** /(E*)±A*) and the norm in ~4(E~ is equal to the norm in A** /(E*)~A*. PROOF - We first give some explanations regarding the formula in the theorem. Since A is dense in E the couple [E, A] has a conjugate couple [E*, A*] with E* C A*. The second conjugate couple may not exist since E* is not necessarily dense in A*. If we consider E *A*, which is a closed subspace of A*, its conjugate (with conjugate norm)is given by A**/(E*)±A*. Now, the couple [E*, E *A*] has a conjugate couple which is [E* *, A**/(E*)±A*]. Obviously, E** D A**/(E*)~-A'). E being a closed subspace of E**, the inter. ¢ section E (5 (A**/(E*) .~-A*) is a closed subspace of the second term and hence, provided with the norm of the second term, is a BANACI~ subspaee of E. To prove our formula, assume first that x eA(E). Then there exists a sequence Ia,}CA such that ]i a,, []A -- ]] X, l].~(~) and []a,,--~c]tz--0. We have then J < ~* * -- a, a~ >.4* l < ~, k ---- 1, ... ,-n In particular, when the al* are in E*, we get I Anna~i eli Matematica 12 90 ~. ARONSZAJ~N - E. GAGLIARDO: I~terpolation spaces, etc. and ]I x ItA(E) < R'. Hence, choosing R' arbitrarily near to R : tlx l!~*)" we get II ::c ~.~(E)I~ _-- < iI x/!~A*), which finishes the proof. [IO.VI] COROLLARY - If A is reflexive, then .~(E) --_ A. PROO~~. - By [10.III, c)] we can replace E by ~E and thus assume that A is dense in E. Then E'CA* and E* is dense in A* (since E* is a total subspace of A* and A is reflexive). It follows that (E*) ±A* ---A*±A*--(0) and ~(R) _~ E G (A* */(0)) -- E N A = A. [t0.VII] REI~IARK - By [10.III, h)] the repetition of the operation << com- pletion rel. E>> does not lead to any new spaces. However, a different kind of iteration of the relative completion may lead to new spaces. Put Ao--E and for n ~0 put A,_~-----A(~). A priori, this sequence can be extended into a transfinite sequence by defining for ordinals ~ of second kind A~---- ~ A~. However, this transfinite sequence must become stationary and the last A~ different from the preceding must be equal A. At present we do not even know if this last index of << non-stationarity )~ may be infinite. Actually we know only of examples where this index is ~ 2. Such an example, with index 2, is constl'ucted as follows: We take A----l~ (bounded sequences converging to zero). Then A**--l °~. For any prime number p consider the element uv = {~,,}eA** defined as follows. For n:p~, l~k~p, ~--1. For n--p* and k>p, ~,=-l/p. For all other indices n, ~,--0. Denote by B the closed subspace of A** generated by elements u~; then ANB---(0) and A--}-B is not a closed subspace of A**. Consider now the HILBERT space H of all sequences {a,, } such that Vn-2[~,,[ ~<~. Clearly HDA**DA. We define E:H/I~ H and identify elements of A with the equivalence classes containing them. (Clearly A N/~H --(0)). Using Theorem [10.V] we prove that the closure of A in A~ is equal to (A-b B)a*'/B, and the last space is strictly greater than A. However, it can be proved that A2 = A, A simple example with index 2 is given by E----L~[0, 1] and A-- C[0, 1]. Then A~--L~[0, 1] and A~--A. Here, however, A is already a closed subspace of A,. [10.VIII] THEOREM - Let [V, W] be a compatible Banach couple. We have a) (V n W) (v+W~ -- (V N W)(V) + (V n W) (w). b) ~(v+v) = V + (V N W) (W), "W (v+r<) "=-- W + (V n'w) (v). e) ?(v+~> ~ ~v Q~) In general the • ~ ,) cannot be replaced by , Z:,. 2(. ARONSZAJN - E. GAGL1ARD0: Interpolation spaces, etc. 91 PROOF. - a) By [!0.III, f)] it is clear that we have the inclusion (~ D >~; hence it remains to prove ((C>>. Take xe(VNW)(V~ -W). There exists a sequence Iu, iCVN W such that Hu,,ilva)v II x~ - ~. Hvaw~ max [( I! x, !, + ~:tv. iI.), (1I x~ lt~ + !l'~v. II,~ + R)]. Hence x~ e (V (3 W)(7) and similarly, xz e (V/"3 W) (W). b) Here again the inclusions ((D ~> are obvious, and we prove (< C ~. Suppose xe~Z(v+Y). Hence there exists a sequence {v,}CV with liv, l.v c) is an immediate consequence of b) and a). [10.IX] THEOttEM - Let [V, W] be a compatible Banach couple and A and B two interpolation spaces behveen V and W and A cB. Then ~t(B) is also an interpolation space between V and W. If A is normalized so is ~(B). PROOF. - Let T~ ~;[V, W] and ace ~t(B). Then there exists (a, }CA such that tl a. ]!~ = II x I/_~'(.> and ]i ~ -- a,, ll---O. It follows that li Ta,, I1~ --<1 T j., 1t x IIZ(.) and li Tx -- T a,, ))B ~ O. Therefore Zx Z ~B>, fl TX i'i£(B) < I TI.~ ti x fIX(B), and ]TIz(, )~ ]TI~. If A is a normalized interpolation space, then with the above notations, ]Ix -- a,]lv+jr ~ 0, and hence ]]x[]r+w- lim]la~,]lv+l~-< i[xll£(,). Thus ~(B) CV + W. On the other hand ~(S) DA~ VI'3W and I T]g(B> <~ I Tla < I TtV, which proves the second assertion of the theorem. [10.X] LE~IMA - Let IV, W] be a compatible Banach couple, h ~ V + W, and A an intermediate space between V and W. a) Suppose that A satisfies one the following two properties: (lO.2a) The topology o[ A is not coarser than the one of W when restricted to Sva w(R), 0 < R < oo. (10.2a') The topology of A is not coarser than the one of V when restricted to Svnw(R), 0 < R < oz. 92 N. ARONSZAJN - E. GAGLIAI¢DO: lnterpolatiol~ spaces, etc. Then if h~ ~zr+ w) or h~ f¥(17+w) respectively, there exists T e ¢gva w[V, W] such that Th ~ A. b) Suppose that A satisfies property (10.2b) The topology of A is not coarser than the one of V-b W when restricted to Sva w(R), 0 < R < ~. The~ if h ~. V(V+w)U l~r(v+W) there exists T e ¢gvn ~[V, W ] such that Th ~ A. PROOF.- a) By symmetry in V and W it is clear.y enough to prove our thesis if (10,2a) holds and h g ~,(17+w). We show first that (10.2a) is equi valent to the following property: (10.3a) There exists a sequence Iu~!C~TA W such that ~l!u~F!vnw-- 1, Nu~ [~w ~ O and ]I u~ ll.~ ~ ~ for some ~ > 0. Obviously if (10.3a) holds, then for any R, 0 < R < o% ~ Ru~ C Svnw(R) 1 and ~Ru,* converges to 0 in W but not in A. Suppose now that (10.2a) holds. Then there exists }x~ } C Svnw(R) such that x, converges to some x'~ Svnw(R) in W but not in A: Because of compatibility of W and A, {x~} is a CAtchY sequence ia W but not in A. It follows that for some sequences of integers converging to ~, ( m~ } and I n~ }, we will have, if we put Yk = x,~ -- x, k, that I/Yh - o, i[ y, it~ --> ~' > 0 and [I Yh Hvn~7 ~ ~ ~- 2R. We cannot have k = 0 since then HYhlIA would ~0 (the topology of A being coarser than the one of V N W). Hence u,*--Yh/llYkIlVNW satisfies (10.3a). We now pass to the proof of a). If for all Te c6vaw TheA, then Th would be a linear mapping of ~VNW into A. By a simple application of the closed graph theorem it is seen that this mapping would have to be bounded. Therefore, in order to prove the statement of the lemma, it is enough to exhibit a sequence IT,,} C ¢G yaW with 1T,*l,g uniformly bounded and ]IT~httA ~. We will construct 7,* as an operator of rank 1, Tax-- Using the sequence {u~}, which exists by (10.3a), we define zn~--[tu, ll~/~u,. Then -P'fo ~ O. Hence for the operator Tnx--< x, fo > z** we have ITn Iv--0, IT.[w - IIP"fo i ]w'llz,*i!w[ -- tl P ¢'f folIw, and IIT,~hll~= Hz,,H~ >_wHu,,[i~~ -oz. Suppose now ~hat h e ~v+~Y. Consider the subspace BC (V-I- W)* of all f's satisfying < h, f > -- 0. B is a closed subspace of (V -[- W)* of co-dimen- sion 1. P'(B) is therefore a subspace of P'((V-{- W)*) of co-dimension at most i. We prove now that (*) P'(B) is not a closed subspace of P'CV + w)*) in the topology induced by V*. In fact, the linear functional ~(f'), f' e P'((V q- W)*), defined as < h, f>v+,+ for any f with P'f= f' is well defined (since for P'f--O, < h, f>- 0 in view of hE VV+w). Since ~(f') is not identically 0 and vanishes on P'(B) it follows that P'(B) is the nullspace of ¢p(f'). If (*) were not true, the nullspace of~ would be closed in the topology of V*, hence in this topology ~ would be continuous. By HAH~-BANAC~ theorem~ ~ could be extended to a linear functional h** E V**. It follows that h** is in the closure of Sv(R) in the weak* topology of V** for R--IIh** fly**. We will show that as a consequence, h would have to be in the closure of S~(R) in the weak topology of ~V-t-W. In fact, for any finite system l f(k)l C(V-t-W)* and any ~> 0 we can find an element, v~ST(R) such that ] Q(b) -- [I [o __ b [l(v+w)" [1 p'(fo _ b)llv" varies continuously with b and since < h, [o __ b >v+T+- -- t, we have "tl fo __ b ii~'+w)* > ~1h IIv+w, and the quotient Q(b), for varying b on the polygonal line, takes all the values between Q(b1) and ~:x~. We consider now the sequence {u~ i existing by (10.3a), and pick up an no such that Ilu,,l~!~? ~ max [2, Q(bl)] for n> no. Then, for n-- 1, 2,..., we choose bn so that Q(b,,)--Ilu~o+~It-~. We put [~ -- f~ -- b,,, and z,~---]]P'(fo~b~,)tl'f:u,~.+,,. Since Q(b,,) ~ 2 and ]] re_ b, II(v+w)* = max [ii p,(fo_ b,)Hv*, ll P"(f° -- b,)]',,~,], we get ]i P"(f°-- b,)]]w* 94 N. ARONSZAJN - E. GAGLIARDO: InterpolatioJ~ spaces, etc. --t]f °- b,,lI¢v+r~),. Also, since 1 = II u~,o+,,livnm = max tllu~+-ilT*, Ilu~o+,~ll,] and [iU,o+,H~v~< 1/2, we get Hu,0+,/lv= 1. It follows, therefore, that for the operator T~x --- < w, f~, > z,, IT~!v-- i T,)t~,-- 1. On the other hand, since Q(b,)-*~, the b,'s cannot be confined to a finite number of segments joining b~ with bk+1, and it follows that IIP'(f°--b,)liy • -+0. Hence Iiz,~i[,~ It p,(fo_ b,)IIv~. ~ co, and our proof of a) is finished. b) To start, we notice the equivalence of (10.2b) to the property (10.3b) There exists a sequence {u,} C V N W such that ilu~Ilvn~.-- 1, ilu,,]]v+~ ~0 and ilu~ll~ ~ for some ~ > O. The proof is completely similar to the one of equivalence between 10.2a) and (10.3a). Next we show that (10.4) Properly (t0.2b) holds if and only if one of the properties (10.2a) and (10.2a') holds. In fact, if either (10.2a)or (10.2a')holds, then, afortiori, (10.2b)holds since the topolog}~ of V+ W, restricted to V A W, is coarser than either of the topologies of V or W. On the other hand, if (10.2b) holds, then (10.3b) holds and we can use the sequence {u~} from (10.3b). The condition Ilu,,liv+s. ~0 means that u~--u,[d-u~", • u',~ and u+,t! in VN W, Iiu,(ttv---0 and 'alu,,"Hr~-~O. Since llu,;ll. +lluo"ll, >llu,,il >, at least one of the sequences {u,~'} and {u,~"} satisfies lim sup Hu,' Ha ~ ~ / 2 or lim sup Hu-" li* ~ ~/2 respectively. If it is the first sequence then (10.2a') holds, and if it is the second, then (I0.2a) holds. The thesis of b) follows now from (10.4) and a). [10.XI] REMARK- The interpolation spaces between V and W which satisfy properties (10.2a) or (10.2a') are of rather special kind ; those introduced in the literature in connection with interpolation methods almost never satisfy these properties (i7). As examples of intermediate spaces satisfying property (10.2a) we mention here that if V A W is not closed in W then any inter- mediate spaceC V satisfies (10.2a). (Clearly if V V~ W is closed in W there are no intermediate subspaces with property (lO.2a)). The proof of Lemma [10.X] actually gives a stronger resnlt: namely, that the operator T of the lemma can be chosen in the BA~CAClg sub-algebra ~o[V, WJ of ~;vnw[v, W] formed by closure in ~ of the class of all ope- ('~) The usual interpolation spaces have the property that/[uIIA--<--cllu[l~-0 Ilu h~- with constants c and 0 independent of u, 0 ~ 0 C 1. ~*. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. 95 rators of finite rank with range contained in VA W. ~o is also an ideal of ~ and is formed by operators which are completely continuous in every interpolation space between V and W. Passing' to the applications of the lemma we note first an immediate corollary. [S0.XII] CO~tOLLAR¥- Under the conditions of Lemma [S0.X] (part a) or part b)) if, in addition, h e A, then A is not an interpolation space. For a compatible BANAC~ couple [y, W] there usually exist infinitely muay couples [V', W'] with V'(3 W'---V(3 W, and V'-{- W'-'V~ W (all the spaces being mutually compatible) For each such couple the class of intermediate spaces is the same. A natural question arises: are the inter- polatiou spaces also the same. The next theorem gives an essentially negative answer to this problem. [t0.XIII] TttEORE~f - Let [V, W] and [V', W'] be two compatible Banach couples with V-~ W : V ~ W', V (3 W-- V' (3 W', neither of them being identical to [V-~- W, V (3 W] or [V (3 W, V ~ W]. If all the interpolation spaces belween V' and W' are interpolation spaces between V and W, then, possibly by interchanging V' and W', we have the following statements: a) if V (~ W is a non-closed subspace in V and W, then (V N W) (v) C V' C ~7(v+w) and (V (3 W)(~ ) C IV' C(¥ (v+w~ b) If V N W is closed in V or W, then (possibly with interchange of V' and W'), V' = V and W' : W. :PRooF. - Since V' and W' are interpolation spaces between V' and W', by our assumption they must also be interpolation spaces between V and W. We consider the two cases. a) Since V (3 W is not closed in iV there exists a sequence t un'}C v(3 w such that llu 'Ii n = S, and lin 'iiw --0. We cannot have and ][ U , ! w • ~ 0 since this would imply Ilu,, ? ]Ivnw 0. Hence we can assume that in one of the spaces V' and/or W' the norms of u,,' are >~>0. If necessary, we can interchange V' and W' so as to have ]l u'~ ]Iv" > ~ > 0. Hence by Corollary [S0XII] V C ~(v+~v). On the other hand, for any such sequence i~yLn't~g['~ W with It un'IIVNW : 1 and :]u~'[[w --* 0 we must have ]lu~'[[~r, ~ 0. Otherwise, for the same reason, W' would be C ~(v+w); hence V + W would beC~ z(v+~), contrary to [S0.IV]. It follows that every CAUCHY sequence in W, bounded in V(3 W must also be a CAuc~¥ sequence in W', and thus 96 ~'. ARONSZASN - E. GAGLIARDO: Interpolation spaces, etc. by compatibility we get W' D (]7 n w) (Tv). Since also V n W is not closed in V, the same argument shows that one of the spaces V', W' must beC ]ir(v+W) and the other D(V n W) (V). The first inclusion cannot be satisfied again by V' since then the second inclusion would be satisfied by W' and we would have, by using Theorem [IO.VIII], V' C ~(v+~) N ~¢(T~+~r) ___ (V' n W) (V+W) = (V n W) (7) ~(V N W) (~)C W', contrary to the hypothesis of our theorem. Hence we must have W'c~V(V+~'~ and V'D(V N W) (V) as stated in a). b) If vn W is closed both in V and in W, then it is closed inV+W and hence closed also in V' and W' (see Lemma [7.I]). As in Remark [7.X], we see then that either V'~V and W'-- W or W'--V and V':W. Suppose now that VN W is closed in V, but non-closed in W (the case when V(3 W is closed in W and non-closed in V is treated similarly by symmetry). We repeat now the argument from the beginning of the proof of a) (possibly interchanging V' and W') and obtain V'C ~(v+T~). Also, as in the proof o[ a) we get W'D(VNW) (W). Since now VN Wv--VN W~-V, we have, by Lemma [7.I], W 7+n" : W and (V N Wy+~r ~_= (V n w) ~r. v' cannot be CW: Wr'+r~ since otherwise V' would be C(V~(VNW) (~:))N W "- (V n W) (W~C W'. By Lemma [7.1Ii] it follows that V'D V and thus VC V'C V-4-(VN W) (w). If for some ve(VN W) (lr)-(Vn W) we had veV', then also we would have vEV'N W'=VAW which is impossible. Hence V'--V. Checking with Theorem [7.IV] we see that W' cannot be in case 1% nor in case 2 ° (since it would follow that (V'N W'D V)); henc~e it must be in case 3 ° or 4 ° . In both these cases we would h~ve W'C WV+~I~-- W and since V'= V, we must have W': W (otherwise V'A-W' would be strictly smaller than V + W). [10.X[V] COROLLARY - Under the hypotheses of Theorem [10.XIII] part a), if (V N W) (v+W) -- V n W (in particular, if v,n w is reflexive) then V' -" V and W': W (or V': W and W': V). PROO~~. - In the present case (VN W) V--(VN W) w-- VN W; hence ~-(v+n:)_-V and ]YV(V+w)=-W (see Theorem [10.VIII]). It follows by the preceding theorem that V'CV and W'CW (or V'CWand W'CV) and the inclusions must actually be equalities because of V + W : V' ~ W'. [i0.XV] REMARK - We do not'know of any example pertaining to the case a) of Theoren [10.XIII] where the couple IV', W'] is not identical with [v, or [w, v]. ~N. AROZCSZAZY - E. GAGLIARDO: Interpolation spaces, etc. 97 [10.XVI] T~EOR]~M - Suppose that neither of the two compatible Banach couples [V, W] and [W, V] is identical to [V N W, V + W]. a) If V N W is a non-closed subspace in V and in W, then neither V nor W are interpolation spaces between V n W and V + W. b) If v n w is closed in V but not in W, then V is never an inter- polation space between V n w and V+ W whereas W is one, if and only if V n W is dense in W. c) If V N W is closed in both V and W, then neither V nor W are interpolation spaces between V n w and V ~- W. PROOF.- a) Suppose V is an interpolation space. There exists a sequence fu,,icvn w such that Ilu. ll~n~=l, Ilu.ll -O and Ilu. llw b) Suppose first that V is an interpolation space between V ('/W and V-{- W. We can then repeat the argument at the beginning of the proof of a) and obtain again a contradiction to [10.IV]. Concerning W, in present circumstances it is a closed snbspace of V-{-W (see Lemma [7.I]). The only interpolation spaces for the couple IV N W, V q- W] which are closed sub- spaces of V-b W are V + W and V n w v+w_ v N W W. Hence our assertion. c) This case was settled in Remark [7.X]. § I1. Minimal interpolation spaces. - in this section [V, W] is a fixed compatible BAZ~ACH couple. [ll.I] TItEOREM - Let A be any normalized intermediate space between V and W. Then a) There exists a minimal (rel. the order relation C) normalized interpolation space ~V(~ ~D A. It is formed by all elements u ~ V -b W repre- sentable by admissible sums E T~a such that T~ ~ ¢g with at most an enume- a~ 4 rable number of T~' s~ O, the sum converges in V-k W to u, and ~ ITa I~g ]1 a li.~ < oo. aGA The norm of u in this space is given by I1 u II Orc~ ~ inf ~=AITal~llalJ'~Z over all admissible sums representing u. Annal~ di Matematiea 13 98 N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. b) There exists a maximal normalized interpolation space, z.r.r.r.r.r.r.r.r.rff~~ A. It is formed by all u~A such that for every T~ ~, Tu@A. The norm in this space is given by JjuJje~q~- sup li TulJ~ for all Te~ +vith J Tj~g< l. PROOF. - The existence of the maximal and minimal interpolation spaces spoken of in a) and b) follows immediately from the completeness of the lattice of normalized interpolation spaces. It remains to prove the explicit representations given in a) and b). a) The proof that the space ~,~ described in a) is a B_A~ACH space is completely similar to the proof in part b) or Theorem [5.XII]. In particular it is a normalized BA~ACIt subspace of V ~ IV, since for any admissible sum /tuJlr%w < E il T~allv+w < E I T~l~g[lallA. If ueA, then the assignment a~A a~ A Tu -- [ (identity), and T a ---- 0 for a ~ u gives an admissible sum for u which show-s that /luj]~!~LA <--]lUi'A. Hence 917~D A D V N W. Furthermore, if Te ~, then for any admissible sum u = E Tan we get Tu --- E TTaa and since EJTT al~gllaHa~jTj~gy,ITalcgljall/ it follows that ~'6.~ is a normalized inter- polation space. Finally~ if B~A is a normalized interpolation space and u e~th:~, then for any admissible sum u--ZTaa we have ~. HTaaNB~ v i Ta ]~ Ha HB ~ Y, IT a J~d I] a [j.~. Hence u E B and ]] u lIB ~ Jl U J!~ihA, which finishes the proof of a). b) In proving that ~A is the maximal normalized interpolation space ~A, we begin with a preliminary remark. Every u e~TqL:~ gives rise to a mapping of ~ into A given by Tu. This mapping is linear, and by using the closed graph theorem one shows imme- diately that it is bounded. Thus we have a correspondence between elements in ~A and ~;A,~. This correspondence is obviously linear, one-one, and with the norm given in !~A it is an isometry. One checks also that in this correspondence the image of ~A is a closed subspace of ~A,~;. Hence ~ is a BANACI~ space. Since Ilu![g~; l = sup JlTu =llull:4> => II-ll, + and since for u E V (3 W, Iluli~'i5 A < sup II .ll n <_ll li n , we see that ~[L~ [T:'~'~_~ 1 is a normalized intermediate space between A and V ('I W. On the other hand, for I T']~ -- l we have ]lT'u J,~Ic~--I ~,~ sup I!TT'uJ]:~ ~ IIu:]~A and thus !~ is a normalized interpolation space. It remains to show that if B is a normalized interpolation space with B~ A, then B ~ ~CA. In fact, if u~B, then TueBCA for all Te~ and H uJ]+=-E~ -~ sup ]1Tu [l~ snp il Tn <- II u il . ~. ARONSZAiIN - E. GAGLIARD0: I~tcrpolation spaces, etc. 99 [tl.II] I)EFI~ITION -- For any us V-b W we denote by ~(u) the set of all ~ ~ Tn with T s ~. The space '~(u) will be considered with the norm ]Ixll,~ --- inf ITI~, the infimum being extended to all Te ~ with Tu -- ~c. [li.III] - For each us V + W, u ~ 0, we have a) ~(u) is an interpolation space between V and W. ~(u) is a nor. realized Banach subspace of V "b W for p _> [t u ]ll~+~v. b) For every T' ~ ~, [ T' I~'(u) <-~ JT' ]~g. c) V A WA ~(u) for every ue V + W. d) ilUi!u=l, if u~V(5 W then ilu--y!iu >-I for every yeVUW. PROO;~~. - a) For fixed u, Tu is a linear mapping of "-~ into V-i- W and C~(u), as defined, is the range of this mapping. One checks immediately that this mapping is continuous. If ~ is the nullspace of this mapping, then by definition of Ilw]Iu themapping becomes an isometric isomorphism of ~/~VLu onto ~(u). Hence ¢~(u)is, a BAleAClt space. Since for each x--Tu'with Ts ¢~, t!xlIv+w<=tTl~;]]utiv+w it is clear that ~(u) is a BA~AClX subspace of V + W and that ~(u) is a normalized BANACE subspace of V-}- W for p ~_> ilultl++~r. That ~(u) is an interpolation space is obvious. Furthermore, if x--~(u) and T'e ~, then IIT'xllu-- inf [TI~- ~ inf [T'TJ~ <_ i T']~;liW[[u. Thus parts a) and b) are proved. T~=T'x Tu=z c) Follows from a). d) If Tu ~- u then I Ti~'=> 1. Since Iu = u and J I[~---- 1 we have t!Uilu-=l, If ug. VA W we have by Corollary [9.V] that for yeVNIV and every Te~ with u -y-- Tu, [ TI%-~1. Hence the second part of d). [II.IV] THEOREM.- Let B ~ (9[u])~A where A is a normalized interpolation space, [u] is the one-dimensional space get, crated by u e V-+- W, u g: O, with the norm of V + W, and p > 1. b) If u s At • A, A1 is a normalized interpolation space, and ~ :,lu ~llv+w=> ii u llA~, then ~DThB~ A~. c) 1[ u A, then Ilu;Ie ,= ilujl +, and Fru -ahem,-> il lIv+, for every a e A. 100 N. ARo~szaJ~x - E. GAGLIARDO: Interpolation spaces, etc. PROOF. - a) We prove first that 01~' = (pllU]lv+w ¢g(u)) ~ A is a nor- malized interpolation space. We ~pply Prop. [6.XIV, e)]. This is possible in view of [ll.III, a), b)], since [1 l.III, a)] implies ~ II u IIr+w ¢g(u) ~ (~ [] u I'7+w ¢g(u)) ~ (V-{- W) for ~ _~ 1. By [ll.III, d)] ~Nul[v+wCg(u)~[u]. Hence ~D~'d'~B. If we now prove statement b) of oar theorem, with ~]~Cs replaced by ~e)iU we will have at the same time proved both a) and b). To this effect we notice first that under the hypothesis of b), Cg(u)CA~ and for w= Tu and T~ ~, It w IIA~--~ I T tog I: u tla~ ~--~ I T I"/7~iI u I!v+w. Hence ~II u !lv+w Cg(u) ~ At which proves the desired statement. c) The proof is analogous to that of [ll.III, d)). [ll.V] COROLlaRY- If A~ and A~ are normalized interpolation spaces, A~ ~A~, and A~ not a closed subspace of A~, then there exist interpolation spaces strictly between A~ and A~; for instance, the spaces ~D~f'Cn where B- (~[u])~A~, ueA~ ~- Az and ~ --Ilutt~,/llulI~+w. PROOf. - Under present conditions, ~[u]CA~; hence B~A2, and A~ ~['5BCA~. Obviously O]'CB @ A~ since u ee.~)~. On the other hand, ~ff'dB cannot be = A~ since, in the topology of ~B, U is not in the closure of A~ (by Theorem [II.IV, c)]) whereas in the topology of A~, u is in the closure of A~. For brevity's sake, we will introduce the notation (lZ.1) 9rc~,~=grc,~=(P~(u))vJ(vnw),l[ IIo.,=11 IlercB, where B = Ilullv+ [u] u(vn w) and 9 ~ By Proposition [ll.III, c)], ~,~, -- ~(u) but the norms on the two spaces are in general different. [ll.VI] THEOREM - Consider an arbitrary set ~ C V -~ W and a function ~(u)___~ llullr+~7 defined for u~. Then the space ~ ~)rCe(~),~ is a normalized ue$ interpolation space between V and W and all such spaces can be obtained in this way. PROOF.- That U!~FC~(,,),~, is a normalized interpolation space follows from Theorem [b.XIII]. Suppose now that A is a normalized interpolation space. Put ~--A--(0) and ~(u)--llulIA for ueA. By Theorem [ll.IV, b)], ~]'Cg~), ~ C A for every u ~ $. Hence t.~ ~'Cpc,,), ~ C A, On the other hand by [II.IV, c)] tlullp(,),~,-'--llull~. Therefore, AC~J~)YCpO,),, and the norm in U ~]7~(~), ~ _~ the norm in A, i.e;, A C ~Ce(~), ,. N. ARO~-SZ~_JN - E. GAGLIARDO:~ Interpolatio~ spaces~ etc. 101 § 12, The structure of the spaces ~(u). - As in the preceding section [V, W] will be a fixed compatible BA~AC~ couple. We will speak about interpolation spaces (meaning interpolation spaces between V and W). The last theorem of section 11 showed the role played by the subspaces Ca(u) (or, more precisely, their normalized versions ~p,~)in the construction of interpolation spaces between V and W. In this section we will collect some more properties of the spaces '~(u). It will be convenient to use the following definition. [12.I] DEFINITION- In V+ W we introduce a partial order relation denoted by <. We write u <~ u~ (or ux ~- u) if ue ¢g(u~), i.e., if ~(u) C ~(ux). If u- [12.II] - a) The following sels are equivalence classes : (0), (V G W) -- (0), (V--VNWV)+(W--VN Ww) (~8), v-Vl'~wv, w--]znww. All the other equivalence classes are contained in one of the five mutually disjoint sets : (V-- V N W v) + (V N W ~-(V N W)), (V N WT--(V N W)) + ( W - V N W,V), (V n W v -- (V N W)), (V n w w -- (v N W)), (V n W W- (V N W)) + (V n W "~- (V n w)~. b) Every interpolation space is a union of equivalence classes. Hence if A and B are interpolation spaces, A C B, then B -- A, i7. B ~ A, and B ~ ~B are unions of equivalences classes. PROOf. - a) The proof of a) is immediate if we use Lemmas [7,1] and [7.III] and Theorem [7.IV]. We have ~(0) -- 0 ; for u ~ (V fl W) -- (0), '~(u) -- VFI ~V; for ue(V--VA Wv)-~-(W -VAWW), ~(u)-- V+ W; for u~V-- V N W v, ~(u):V; for ue W~ V fi W w, ~(u)-- W. b) The proof is obvious since under the hypothesis of part b) ~s is an interpolation space. [12.III] - a) If T~ and T -~ exists in "~ then Tu~x.u for all u. b) If xe ~(u) and I]x-uHu < 1, then x-,-u. c) The equivalence class of u forms an open set in Ca(u). [is) For two subsets A and B of a vector space, A + B means the set of ail vectors of the form a-~ b with a ~ A and b q B. If one of the sets A and 13 is empty, then A-l-B is also empty. 102 X A~o~sz.~z~" - E. GAGLIARD0: Interpolatio~ sp,aces~ etc. PROOf. - a) is obvious. To prove b) we notice that if iix--ulI~ < I, i.e., there exists Te~ such that ITl'~ [12.IV] - a) Each interpolation space A C ~(u)- (u) lies outside of the open unit sphere S~(u, 1) with center u. b) If u~ ~ ~(u) and u~ 4 u, then there exists a maximal interpolation space A~:, (in general not unique) with u~ e Am~:, C ~(u) -- (u). Am~x is always a closed subspace of ~(u) contained in ~(u)- S~(u, 1). c) For every interpolation space A C ~(u)--(u) there exists a linear functional c?(x) on "~(u) with bound 1 such that ¢?(u)- 1 and ¢?(A)~-O. The subspace ~(x)~ 0 is a supporting hyperplane for S~(u, 1). PROOF. - a) If u ~ A then A does not contain any point of the equi- valence class of u hence [12.III, b)] implies Afl S~(u, 1)- 0. b) There exists at least one interpolation space, namely ~(u~) satisfying u~ ~ A C '~(u) -- (u). By a), all such spaces lie in ~(u) -- Su(u, 1). Hence .~;~u) is also an interpolation space satisfying this condition. For any class of interpolation spaces ordered increasingly by inclusion such that each space is C ~(u) -- Su(u, 1), the closure (in ~(u)) of their union is again an interpo. lation space C ~(u)- Su(u, 1). Hence the conclusion of b). c) It is enough to construct the functional ~ for the closure 2~"~tu) C ~(u)--Su(u, 1). We define it first on the closed snbspaee A ~(u) ~ [u] by putting ~(a -}- ~u)--~ where a e ~:(u). Since a ~ u, we h~ve for ~ ~0, II~ T ~u Ilu---- I~I II~ -~a + Ul[u ~ t~l and thus q~ satisfies our requirements on A~(u)~q --In]. Using HAHz~-BA~cACrI Theorem we extend q~ to the whole of ~(u) according to the requirements. [12.V] REMARK - There is an interesting algebraic interpretation of the last proposition. Consider u ~ 0 and denote by ~, the set of T e c~ with Tu = 0 (the nutlspace of the mapping T ~ Tu). Then ~, is a closed left-sided ideal of cG and there is a one-one correspondence between interpolation spaces A C ~(u) and left-sided ideals ~9~,,A satisfying ~C~7~,~C which are BA~AC~ subspaces of ~. This correspondence is given by ~,,,~-- [Te ~:TueA]. Part b) of [12.IV] shows that there are such maximal proper ideals and that they are necessarily closed in ¢g, The functional ~ of part c) N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. t03 gives rise to a bounded linear functional ~(Tu) on ~ which, on the subat- gebra [I] + ~Y'5,,,_~ is multiplicative. An especially interesting case is when A is a closed subspace of ~(u) of eodimension 1. Then ~,,,A is a two-sided maximal proper ideal of "-~ and ¢p(Tu) is multiplicative on the whole of ~. In the remainder of this section we wil] investigate the possibility of the extreme case mentioned at the end of the preceding remark. We actually study the more general case where (1,,.1) ¢g(u)C [u] + A, A being an interpolation space not containing u. If (17.1) holds, then obviously C~(u)--~[u]-[-(A N ~(u)) which presents the ease mentioned at the end of Remark [12.V]. [12.VI] - Let A be an interpolation space satisfying (10.2a) or (10.2a') or (10.2b). Then, in order that relation (12.1) hold, it is necessary that u e ~V(v+~r) or, u~ ~z(v+w), or, u~ (Z(v+v) U ~¢(v+~) respectivelj. The proof is immediate by Lemma [IO.X] (the space A in the lemma being replaced by [u] -k- A). [12.VII] - If the relation (17.1) holds, then: a) If A satisfies (t0.2a) and (102a'), then ueVfl W(V+w). b) If u ~ V fl W (v+W) then A satisfies (10.2b). PRooF. - a) By the preceding proposition uE ~(v+w) N VV(v+w)"-V N W(V+w) (see Theorem [10.VIII, c)]. b) If A did not have property (10.2b) the topology of V+ W would be finer than that of A on every sphere Svow(R), 0 < R [12.VIII] COROLLARY - Suppose V fi W is non-closed in V and W. Then the relation (12.1') ~(u) -- [u] + (V fl W), u~VNW is possible only if u ~ V fl W(V+ W) 104 N. AROIqSzAhR~ - ]~. GAGLIAltDO: Interpolation spaces, etc. PROOF. - In the present case there exists a sequence t u,~t C V N W with llUnII~ow--t and Ilu.ltw~0. Hence VN W has property (10.2a)(also with V and W interchanged). We then apply the preceding Proposition [12.VII, a)]. [12.IX] REMARK- The ease when (12.i') holds represents the smallest spaces "g(u) after the space (0) corresponding to u -- 0, and V N W correspon- ding to u~(VN W)--(0). If, in the corollary~ we did not suppose that VN W is non-closed in V and W, the only additional possible cases of (12.1') would be the trivial ones when V -" [u] -~ (V N W) or W-- [u]-t-(VN W). When (12.1') is valid we can also completely determine the equivalence class of u: it is the set of all elements of the form ).u+~ where xeVN W and ),~0 (since V N W is then the only maximal interpolation space C ~(u)- (u)). As a counterpart to the corollary we construct an example where u e VN li '(r+W) and where the relation (12.1') actually holds. Let V be an infinite dimensional BA~ACH space and let v,, ~u ~0 in V. We may and will assume that the elements u and vn, n--1, are linearly independent and that ltull{-~ ltv.Iv--1. We construct the elements uh--v,~, k--1, 2,..., by induction, choosing u~--v~. If ~all u~...ua--v,~ are already defined~ we consider the smallest distance ~h from u to the linear subspace [u~,..., uk] generated by u~,..., uk. We denote then by nk+~ the smallest integer >n~ such that Iiv,b+ --ull~-~2-a~a. The sequence {uh! OO being so defined, we consider the subspaee W~ (~ fuji and the linear couple IV, W]. We notice first that u ~ W. Otherwise we would have u -- Z ~kua with Z ]~h] ]IuhIIv < ~. Since IlukNr-~ I[uIIT~- 1, we may find N such that Z l~ht It follows that Z I~al ~ M and I al ~ M. If we put x'-- Z ~ua Jr (~¢ -- }2~a)u, i 1 1 then x':x. In fact, x, -- ~'-- Z (~')--i~)u~+ Z (~')--ia)(ua--u)+ k=l k=N+l N u[ Z (ia--i(~"))+( Z ~(~'~)-- :¢)]. When n---~ and N remains fixed we get N. ARONSZA;IN - E. GAGLIARDO: Interpolation spaces, etc. 105 lim sup ]Ix,,- ~c'll~ ~ 2M2--N~N. Since we can take 1V--~ ~x~, this shows that x, ~w' in V; hence w'=x. It follows that ~7(v) C[u]+W. 0n the other hand, Ilu,]lw~ 1 and u, ~ u in V; hence u~ !~V(v), and our proof is finished. C~AV~E~ III Interpolation Methods. § 13. Interpolation theorems. - In this chapter << couples >> will mean BA~AOE couples (unless otherwise stated). Different couples which we con- sider will be distinguished by lower indices such as [Vo, Wo], [Vx, Wx], IV,, W~]. We already introduced the notation ~S,a to denote the BANAC~ space of bounded linear transformations of A into B with norm --ITIB,,~ for any such transformation T. for two couples IVy, Wi] and [Vj, W j] we can consider the mappings of V~ U Wi into Vj U Wj which transform V, into V~ and W~ into Wj linearly and boundedly. As we know from Chapter I, they can be extended in a unique way to a bounded linear mapping of V, + W~ into V j + Wj. This class of mappings, called bounded linear mappings of couple ~< i>> into couple < (13.1) IT]i,~- max[ITtvj, v, ' tTIwj, wi]. If T e '~2~ and T' ~ ¢g82 we can compose these tranformations and obtain T'T ~ ¢Gsl. We note first a few obvious properties of the above introduced ~otions. [13.1] - a) ~i,i is a Banach space with ]TI~,i as norm b) If Te ~o,1 and T' ~ ~l,e then TT' e ~o,2 and ITT'1o,2 ~ IT[o,1 IT'11,2. c) The space ~, i can be canonically identified with a closed subspace of ¢gv i, v~ -~ ~yi, w i composed of couples i T', T'} satisfying T'u i :- T"u i e V~ ~ W~ for all uj e Vj N Wj. This identification preserves norms if, on the direct sum, we take the maw-norm. Clearly, part c) above is an extension of [6.1V]. As an immediate exten. sion of [6.I] we obtain [13.II] - For Te ~i,i, [TEv;n~,,vjn,~j < [Tki, [Tlv+~ < ITkj. Annali di Matematiea ~ 106 N'. ARONSZASN - E. GAGLIARDO: Interpolation spaces, etc. [13.III] DEFINITION - [f in a couple [ri, Wi] we choose an intermediate space A~, the system IVy, W~; Ai] will be called a triple, and if no confusion arises, it will be distinguished by the same index as the couple (~9). If the intermediate space is normalized, we will call the triple normalized. If for two triples • i >> and ( [13.IV] TEEOREM - a) If for two triples < b) If the (i ~j)-interpolation theorem holds, then there exists a con- stant c such that I T tA i, ~ ~ C t T {j,~. Consider a triple IV1, Wi, Ai]. Since V i N Wj c A i c V i + W i, there exist constants mj ~ m1(A ~) and m i' ~ mj(Ai) such that (13.2) ]la]]~j ~ mtl]allv~+w j for aeAi, (13.2') llull~ <= m/liutiv~n,:~ for u e v i n w i We will assume that these constants m i and m i' are the best possible in inequa~ties (13.2) and (13.2') respectively. Consider furthermore a couple [V~, W~]. We will use the following notations. (13.3) For any T~,ie ~,i and ueTi,~(Ai), Hu [r ,s(Ap-= inf[ allA j, the infi" mum extended over all a e A i with T,,ia--u. (13.4) For any Ti,~e cgi,~ and ue T~,~(Ai) (so), IlU[!T/~(,tj ) __ max [mi'[luli~ +w+, 11Tj,'uHAj]" (i~) If for the same couple two differont intermediate spaces are chosen, the resulting triples are different and will have to be indicated fully~ say [V~, ~7t, Al] and [Vi, W~, Bi]. (~o) T~,~(A]) is the subspace of all elements u e V¢+W~ such that Ti,iu e Aj. N. ARONSZAJN - E. GAGLIARDO: Interpolations, spaces, etc. 107 [13.V] LEM~A- a) For each T~,¢e ~;~,i with ITi,¢li, i ~ l, the space Ti,i(Ai) with norm (13.3) is a Banach subspace of Vi ~ Wi salisfyin9 Ti,i(Ai) mi(V~ + Wd. b) For every Ti,~ ~,~ with I T~,~li,i ~ 1, T~(Ai) with the norm (13.4) is an intermediate space between V~ and W~, and we have m/(V~ W~)~ • ~,~(A~) ~mi'(V~ + W~). PnOOF. - a) That T~,i(AI) is a BA~AC~ space is clear since it is cano- nically isometrically isomorphic to the quotient space A¢/N where N is the intersection of the nullspace of T~,i with A1 (N is closed in A¢). For u e TLi(Ai) and a~Ai, with T~,ia-- u we have, by using [13.II], II uJI~+~ IT~,II~,i 1 • l[ a lvj+w~ _-- b) To see that T~,~(Ai) is a BA~AC~ space we consider the direct sum m/(V~ + W~)-~ At, with the max.-norm and notice .that the correspondence u ~ {u, Ti,,u! is an isometric isomorphism of T~(A/) onto a closed subspace of the direct sum. Thus the inclusion T~,~(Ai)~m/(V~-4- W~) follows imme- diately. To prove the other inclusion, consider u e V~ f3 W~. Then Ti,~u e V i ~ W~ and II T],iutl,~y < m/ [I TI,iUHV]f~Wj ~----m/[Ti,~[¢,~ [I Ultvin ~ ~- m/ N ullr'~nw, (using [13.II]). This implies T~(Ai) ~ m/ (V~ ~ W~). [13.VIi LEMMA- a) For each u eVi(~ W~ and for each ~ > O, there exists a transformation T~,]e ~i,~ with IT~,~l~,i--1 of the form T~,ix~-- 1 b) For each u~ Vi + Wi and for each ~ >0, there exists a transformation 1 u' Tj,~e~i,i with bound ITj,i[i,i--1 of the form Ti,ix,= <~,f>v¢+w~ where f~(V~+ Wi)*, u' e Vj(] W~ and ~ > O, such that llTl,~ull.~j>~_ (m~ -- ~) )) u )!v,+~,. PRoo)'. - a) Since the constant m i in (13.2) is the best possible, we can find an aoeA i with I[ao]lAj <(mi+~)llao]lvj+wj=mi-]-s. We choose f~(V i + Wj)* so that [I fll(vj+wj)* -- 1 and < no, f>vj+w~ -- 1. Since I T~,i[~,i-- 1 --a max[ H ftIvj* tt utIv,, IJ fllwj* tt ull)~], by choosing ~.---- max[ tl fJI%" tl uHv,, II flJ%" Hu ti~] we get I T~,i [~,i -- i. It follows that T~,i(aao )- u and tluilTi,i(.~i) _~< [taaoii~tj < @mi + ~) ~-- (m i + e) il u tiv, flw, since :¢ --< max ( II u iiv,, t[ u ltw~)" max ( [] f l[v,', I[ f Hw.¢*)= tlu [Iv, n ~ II f II<~/+wj>* = II u liven ~', (we use here [8.I]). 108 N. AaoNszna~ - E. GAGLIARD0: Interpolation space% etc. b) Similarly as in a), we choose fe(Vi+ W~)* with tl f ll(v~+~i)" = 1 and < u, f >vi+w = ]] Utlv~+w i. We choose u' ~ ViA W i so as to have [] u' I!,~i> (m/-- ~)]] u'[[~-in~[.. By choosing then ~ = max [!l/live* I] u' II~j, ]lf I[w." II u' iiwj] we 1 obtain ! Ti,i Ii,i = 1, t[ T~,iu 11~ -- II u tl u ~+ u~ -~ N u' tl~ >= (mi' -- ~) ]] u liv ~+w~ . [13.VIII TKEORE~I - a) The space (13.5) is an interpolation space between V~ and Wi with property (6.3) (i.e., I TBtit "<: ITli,~ for T~ 'g~,~), and satisfies (13.5') mi(V, + W,) ~ B,~ ~ m i (V~ fl W~). b) The space (13.6) C~ =-- C~i (A i) (~ T~ (Ai) is an interpolation space between V~ and W ~ with property (6.3) and satisfies (13.6') b') C~i(Ai) is the space of all u~ V~ + Wi such that for every Tj,ie ~i,~, Tj,~ u E Aj and such that sup I[ Ti,i u IIA~ < co. For u ~ C~i (Aj) we have also I1 u llcj(~ i) = sup tl ri,,u ll.~i" I ~j,i li,~ ~ PROOF. - a) By Lemma [13.V, a)] and Theorem [5.XII, b)], B~i, as given by (13.5), exists and satisfies B~Cmj(V~+ Wi). The second inclusion in (13.5') follows from Lemma [13.VI, a)]. To prove that B~i is an interpolation space with property (6.3), consider any u eBb1 and Ye ¢g~,i with iTl~,~<=l. Then by definition of ~, u--Z T~,}a,, the sum converging in V~q--W~, T(.") I~,j < 1, a, ~ A i anti ~. II an ]l~i <~. Then Tu = E TT~ a, ; clearly, the sum¢71 converges= in V~+ Wi and ITT~:}I~,i~-1. It follows that TueB~t and ]l Tu ttB~ =< tl u tlmi which proves property (6.3). b) That C~i is an intermediate space and satisfies (13.6')is a direct consequence of Lemma [13.V, b] and Theorem [5.XII~ a)]. That it is an inter- N. ARONSZA3N - E. GAGLIARDO: D~terpolation spaces, etc. 109 potation space with property (6.3) is based on the definition of i~. In fact, for u~C~1 we have Ti,~u~A¢for all T~,i~y,~ and IIullc~i - sup max- [m/l[ u I[v,+~, [] Ti,~u ]]~j]. Hence for Te '~,~ with ]T[i,~ ~ 1, we get Ti,~TuEA i and [I Tu [[cii : sup max Ira/ II Tu I[v¢+n~, I[ Ti,,Tu ]l:lj] < ]l u ]Ic,1. We use ITi,iij,i<= z here the fact that V~--}- W ~ satisfies property (6.3). b') By definition of ~ (see (5.3)), u E Cii(Aj) if for every Ti, i E '/;i,~, Ti,iu E A] and 1] u [[c~i(.~j) "-- sup max [m/ [[ u []~¢+w¢, [[ Ti,~u [!aft < c~. Since I Ti,ilj,i~-- <~ by Lemma _[13.VI, b)] there exists a T~,, with ]T;,,ti,,---1 and llT~,~ull.~i>_~- (m/-- ~) l] u !l~,+w, for any a > 0, it follows that sup max [m/It u ttv,+w,, HTi,,ut]~j]-" sup t[ T~,~u ]l~j which finishes the proof. [13.VIII] - Suppose that in the triple [ V i, Wj ; A~] we replace A i by another intermediate space Ai ~ PAi, for some p > O. Then. a) The constants m i and m~' corresponding to ~4i in (13.2) and (13.2') satisfy m t <-~ pmj and m i' ~ pml . b) B~i(Aj)~pB~i(Ai) and CiJ(Ai) D pB~Y(Ai). PROOF. - a) For any aEA i we have l} all,4/>- 1/p II a II~j ~ ~j/~ lI a iivs+~:+ and for any uE ViNW i we have [[utlXj~ P]]uH~j~Pm i'l[u[Ivifl.~. Since m] and mY' are supposed to be the best possible constants in their respective inequalities, our conclusion follows. b) We check immediately that TCi(3.i)~P(Tci(Ai) ) for every T~,jE '~i,i and also T~,~(A~)~p(T~,~(Ai) ) for every TI,iE ¢gi,i" For the last inclusion we use the fact proved in part a) of this proposition that m/_< pmi'. The con- clusion of part b) follows then immediately. [13.IX] - The following relations always hold: B~ ( C~i (A~)) G A i G Cj~ (B/ (Ai) ) . PROOF. - Suppose first that u E BI~(Gii(Ai)). By (13.5), u is representable by an admissible sum in V i + Wj : u = E T('~)xy,i n with [Y(.").li,~1, ~ :< 1, x,,6 C~J(Aj) and [[ u [[Bi~(C~J(Ai)) -- inf E If X,~[[c~i(Aj), the infimum being extended over all such admissible sums. Since x,,E Cd(A~) it follows that- T(~)xi, i n E A i and 110 ~. ARONSZAJN - E. C{AGLIARDO: Interpolatio~ spaces, etc. ttx,,~llc~(~[i) >= [I T}~,lx,~ II,~" i~lence for each admissible sum, E ~(~)x,,,~ converges in A i to' u and [Iu lIAr. -<--- Z II T(n)x~i,~ ll~j =< ~' [I x~ [Ic~i(~./), which proves the first inclusion of our proposition. To prove the second inclusion suppose u E A i. Then for every T~,iE '~i,~ with ~Ti,iIi,i -< 1 we have T~,iu E Ti,i(Ai) cB~i(Ai). Hence u E T~,~(Bj(Ai) ) and, by Theorem [13.VII, b')], ]tu I]cii(~i¢%>) = sup l]T~, i u t!Bii(Aj) ~-~ sup i] T~,i u IITt, j(Aj) tfUtI~¢ which finishes the proof. In the next theorem we will use the following definition. [13.X] DEF~I~IO~¢ - The best constant c in the inequality of Theorem [1B.IV, b)] will be called the (i ~ i)-interpolation constant and denoted by y~,i. [13.XI] Ti~EOREM - Let [V~, Wi; Ad and [Vi, Wi ; Ai] be two triples with all the spaces fixed except for Ai. a) In order that the (j ~ i)-interpolatio~ theorem hold it is necessary and sufficient that A~ B~(AI). If the theorem holds and Ti,i is the (j-~ i)- interpolation constant then A~ ~ y ~, i B~i (A1). b) In order that the (i--~j)-interpolation theorem hold it is necessary and sufficient that Ai ~ C~i(Ai). If the theorem holds and T1,~ is the correspon- ding constant then Ti,~ A~ C C~I (At). c) In order that there ex~ist an Ai such that the (i .e-v, j) interpolation theorem hold, it is necessary and sufficient that Bj(Aj)c C,J(Aj). If this con- dition is satisfied, then the bilateral interpolation theorem holds if and Only if B~i (Ai) c Ai ~ Cii (Ai). If the bilateral theorem holds, then Yi,i BiI(Ai) C Ai 1 C~i ( A ~). Yi,~ PROOF. - a) If A, DB,i(Aj) then Ti,j(Ai)c BiI(Aj)c At. Hence the (}~i)- interpolation holds. Assume now that the (j ~ i) interpolation holds. Then I T*,il~,,.~i < Yi,i! T*,JI~,i" It follows for u E Ti,j(Aj) and tTi,/li,j <~ 1 that [l u ll.~,--< Yi,~ [t a{t-~j for all a E Aj with Ti,ja- u. Hence A,D yi,j(T~,i(Aj)) and by (13.5) Ai~'(i,IBj(Ai) which finishes the proof. b) Again, A,c C~i(Ai) implies the (i ~j)-interpolation theorem. Suppose now that the (i--.-j)-interpolation theorem holds. Then for all Ti,~E ¢gj,~, ]T~,~I~j,A ~_~7i,~]T~,ili,~, and, for u EA~, Tj,~u E A i. Hence by Theorem [13.VII, b')], tl u [[C~J(Aj) ----- sup IE Ti,~ u ]]_~j < T~,~ II u Ii.~; which proves part b). ITi,~Lt,~ <1 c) is obtained by combining a) and b). N. Aao~szx3~ - E. GAGLIARDO: Interpolatlou spaces, etc. 1il [13.XlI] DEFI~ITIO:~ - If in the triple IVy, W~; Ai] A~ is an interpolation space between V~ and W~, the triple will be called an interpolation triple. If A, is a normalized interpolation space, we have a normalized interpola- tion triple. [13.XIII] DE~]~IO:~ - Let [V~, W~; As] and IV1, Wi; Ai] be two triples with (i -~j)-interpolation theorem. The interpolation theorem is called optimal if we cannot strictly increase A~ nor strictly decrease A i without, losing the theorem. [13.XIV] Tt:IEOREM - Suppose that the (j--+ i)-interpolation theorem holds. a) If the interpolation theorem is optimal, the triples (< i >> and ~j must be interpolation triples. b) In the triple (< i >~ if we replace Ai by B~(Aj) or B~i(Cj~(AI)); and in the triple ~j>> we replace A i by Ci~(B~i(Ai) ) or Ci~(Ai)respectively, then, between the resulting triples we get an optimal (j ~ i)-interpotation theorem. c) If, in the triple ~i)>, A~ is replaced by ~_~cAi, and in ~j>> A i is replaced by A1 ~ Ai and an optimal (] ~ i)-interpolation theorem is thus obtained, then Bii(Ai) c A~ c Bi~(Ci~(A~)), and Ci~(B~i(A~)) ~ zt~ ~ Ci~(A~). PROOF. - a) By Theorem !13.X[,a), b)] it is clear that if the interpola- tion theorem is optimal then A~--B~I(Ai) and Aj--Cj~(Ai). By Theorem [13.VIII it follows that the triples are interpolation triples. b) By Theorem [13.XI] the (j ~/)-interpolation theorem is maintained if I A~, Ail are successively replaced by I Bj(Ai) , Ai } and by {Bj(Ai) , Ci~(B~J(Ai)) !. Since A~DB~i(Ai)(by Theorem [I3.XI, a)] and Aic Cii(Bii(Ai))(by [13.IX]), one sees immediately that in replacing t Ai, Ajt by tB~J(Ai), Ci~(B~J(Ai))I an optimal (j ~/)-interpolation theorem is obtained. Similarly, when replacing !Ai, Ajt by tAi, Ci~(A~)t and then by tB~i(Ci~(A~)), C~(A~)!, we obtain again an optimal interpolation theorem. c) In present conditions we have a (j ~/)-interpolation theorem when replacing tA~,Ai} by tA~,Aj} or iA~,At}. It follows by Theorem [13.XI] that A_IDB~I(AI) and A-jc CI~(A~). The other inclusions follow from Proposi- tion [13.IX]. [13.XV] REMARK- The optimal interpolation theorems which seem to be the most advantageous ones, were not, as far as we know, investigated in the literature. On the other hand, it is known that some of the most used 112 N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. interpolation theorems, such as the M. RtEsz convexity theorem, are not optimal (~). [13.XVI] T~Eom~ - If [VI, W i ; A]] is an interpolation triple and [V~, W~] an arbitrary couple, then we have (13.7) ";i,~ B,~ (Ai) ~ C~(A~) . If the triple ( PROOF.- Consider any T~,iE ~i,j and Tj,~E ~j,~ with ]T~,il~,i~l and ITi,~lj,i~I. Take any uET~,j(A~) and any a~A i with u--T~,]a. Then II u IIT~,~(,~i)= inf ]1 a II,~" On the other hand, Ti,~u=Ti,iT~,ia. Since Ti,~T~,~ ~i,~, and A~ is an interpolation space, we have Tt,~ ~ A i and sup It Ti,~u tt.~-- }Ti,iIi,~<---x sup ]l Ti,iT~,ia I!A~ ~- "(i,1 [[ a fiat.. Hence I[ u Ile~i(~j) ~ V~,~ ][ u [iT~,~(.~j), C~(Ai) yi,i(T~,i(A])) and (13.7) follows. If A i is a normalized interpolation space then ~],i-- 1, and our assertion follows from (13.7). [13.XVII] Tt~[EORE51 - Consider two triples <~ i >> and c j ~>. a) The optimal (] ~i)-interpolation theorem holds if and only if Ai = Bii(A i) and A i -- Cj~(Ai). b) If the optimal (j ~ i)-interpolation theorem holds, then the (i ~j)- interpolation theorem is ~valid also. PROOF. - The proof of a) follows immediately from Theorem [13.XI, a)b)]. For the proof of b), we notice that by a) A i = Ci~(B~Y(Ai)). Since B~i(AI) is an interpolation space between V~ and Wi, Theorem [13.XVI] applies, and BI~(Bj(Aj))cCi~(B~i(Aj)). Hence, by Theorem [13.XI, e)], the bilateral (i <-~j)- interpolation theorem holds. In general the spaces B~J(Ai) and C~i(A]) are not normalized even when A i is a normalized interpolation space in [Vi, Wj]. We therefore introduce the following spaces: (13.8) ~i(Aj) ~ [B~i(A~) (~ (V~ + W~)J U (V~ n W~), (13.9) C~J(Ai) ~ [C~i (Ai) ~I ( V~ Jr W~)J ~J ( Vi N W~) . (~t) These results follow, for instance, from E. GAGLtARDO~S investigation of special interpolation methods [5], :N. ARONSZAJN - ]~. GAGLIARD0: Interpolatgon spaces~ etc. 113 Translating the previously proved properties of B~i and C~i, we obtain the following list of properties for the spaces (13.8) and (13.9). [13.XVIII] - a) B~i(At) -- B~i(Ai), ~i(Ai) -- C~i(Ai), ~l(al) and ~(Ai) are normalized interpolation spaces. b) If A i is a normalized intermediate space, Bii(Ci~(Aj))C Ai ~ Ct~(B~i(A~)). c) If A i is an interpolation space, then (13.7) is valid when C and B are replaced by C and B. d) When A i is a normalized interpolation space then B~i(A~) and ~i(A~) are normalized interpolation spaces in [ V~, W~] satisfying ~t (A]) ~ Bii(Aj). All these statements are immediate consequences of preceding theorems. In some cases we use Proposition [6.XIV, c)] § 14. Interpolation methods. [14.I] DEFI~ITIO~ - Consider a class ~ of compatible BANACH couples. If un intermediate space F[V, W] -- A~ is assigned to each couple [V, W] E ~, we say that F is an interpolation method defined on g when the following condition holds: For any two couples [V1, W1], [V2, W2] in ~, the (1 ~ 2)- interpolation theorem holds for the corresponding triples [V1, W1; A~] and IV2, W2; A2] (where Ai)--F[V~, W~]). If all the interpolation constants are uniformly bounded for all choices of the couples, the interpolation method F will be cailed uniform. If all these constants are _--~1, the interpolation method F is called normalized. As obvious consequences of the definition we have the following properties. [14.II] - Let F be an interpolation method defined on a class of couples gg. a) If [V, W]E~ and A- F[V, W], then A is an interpolation space between V and W. b) For any two couples in ~ the corresponding triples determined by F admit of a bilateral interpolation theorem. PROOF. - For a) we take the two couples of the definition as -- [V, W], and for b) we interchange the couples of the definition. Annali dt Matemattca 15 114 N. A~ONSZAJN - E. GA~L]'AI~DO: Interpolation spaces, etc. [14.III] RE~AI~K - Wo do not know of any interpolation methods in the literature which are not uniform; most of them are normalized. We will show in the next theorem that any uniform method can be made normalized by a su~itaBle ch:ange :-of norm in the:spaces ;~F[V, W].: On the other hand we do not know if every interpolation metlmd can be rendered normalized in this way. The importance of interpolation methods lies in the interpolation theo- rems they give rise to. One could imagine more general ways of obtaining interpolation theorems. For-instance, one might have, in two different classes of couples, ~ an d c~', two assignments of intermediate spaces, F[V, W]-- A and F'[V', W'] -- A' so that for any couple IV, W]~ df and [V'W'] ff OX' there would be an interpolation theorem between the corresponding triples. A number of properties could be obtained for such << generalized >> methods. However, past experience shows that all the interpolation theorems in use are obtained from interpolation methods as defined in [14.I]. We will therefore confine otu'selves to this kind of methods. [14.IV] THEOREM - Let F be a uniform interpolation method with all interpolation-constants bounded by ~ < ~x~. Consider any couple IVy, W ~] ~ and put, for u ~ A~, (14. I ) II u rI:a~ ~ sup sup II Tj,~ u I]~tj where < PROOF, - It is clear that (14.1) defines a norm on Ai. If we take ( [14.V] REMARK -- If F is a normalized interpolation method we have, in particular, that for every [l~, W~J E J~, A~--/J'[V~, W~ is an interpolation space between V~ and W~ with the corresponding interpolation-constant yi,~ ~ 1 (it must then be actually--1). In general, these interpolation spaces will riot be n:ormalized, but if we wish, we may Teplace the norm in Ai by the transformation [A~(V~q-W~)]U)(V~N W~:). This transformation does not change the SPaCe but' only the norm, making A~ into a normalized interpo- lation space. It is immediately proved that with this change of norm F remains a normalized interpolation method. N. kao~sz~z~ - E. G~aDo: Interpolation spaces, etc. 115 [14.VI] BI~I~ W~EO~E~ - Consider a no, malized interpolation method F defined on a class of couples ~ and let ~ be a larger class of couples. There are normalized interpolation methods F defined on ~ which are extensions of F. Among these there are two extreme, F' and F" such that for any normalized extension [f of F and any couple [17o, Wo]E ~ we have ~'[ Vo, Wo] ~ ~[ Vo, Wo] ~ ~"[ Vo, Wo]. The two extreme extensions F' and if" are given by the following formulas (14.2') F'[Vo, Wo] ~ @ B*o(A~), $ (14.2") F"[ vo, Wo] ~ (~ C~(A,), i where (~ i ~) runs through all the couples of the class c~ and Ai- F[F,, W,]. PRoo:~. - We show first that for [Vo, Wo] E ~ and any two couples ( (14.3) B~(A~) ~ C~o (Ai). In fact, take u E To,i(A~) for any To,~E ~o,~ with I To,~Io,~ 1. Consider any ai E Ai with To, ~ -- u ; then, II u ]lTo,i(~) -- inf ]1 a~ 11.4t. On the other hand, eonsi- a i der any Tj, o E ~j,o with lT],oli, o<_~ 1. Then, T¢,ou----T~,oTo, ia~ and I T/,oTo,~IA3.,~; I Ti, oTo,~li,~<~ 1 (since the method F is normalized). We have I] Ti, oU lIAr. ~--< ]] a~ ]IA; and thus II u ]]C0i(A~,) -- sup [[ T¢,oU !l~.~. ~ II a~ II.~ and finally, [Iu I[coJ(.Aj) I~,o Ii,0=<~ iI U Ilro,~(-~.)" Therefore Cio (Ai) ~ To,~(A~) and (14.3) follows. From (14.3) we deduce P'[Vo, Wo]~ F"[yo, Wo]. If ~ is any normalized extension of ix/', then by Theorem [13,X[, c)] we have B~o(A,)c/~[Vo, Wo]~ C~o(A~) for every [V~, W~E c~. We obtain thus, by (14.2') and (14.2"), F'[Vo, Wo]~F[Vo, Wo]~F"LVo, Wo]. It remains to show that F' and F" are normalized extensions on ~ of the method F. We notice first that for [Vo, ~]e~ we have. by Theorem [13.XI, c)], B~o (A~) C Ao C C~o(A~) for every [V~, VV~]E c~. Therefore, since for an inter- polation space Ao with constant To,o~--~l we have B°o(Ao)~_Ao~C°o(Ao), it follows by (14.2') and (14.2") that F'[Vo, Wo]~Ao ~F'~[V0, Wo]. 116 N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. Consider now any two couples [Vo, Wo] and [Vi, W~] in e~. Let T~,o E ~,o with I T~,o i~,o-~ 1. Consider first the method _~'. If uEF'[Vo, Wo] this means that there exist (by (14.2') and by (13.5))admissible sums u--E T("). ai with T('*! o~ <_1 0~$~, 01~ ~ ~ ~- and a~ E A~ = FIVe, Wi,~] such that ~ [I a~ !i~ < ~" Further, I[ u [l~,~vo,wo]= inf'. ][ ai,~lla¢,~ for all such admissible sums. Consider now Tx,ou. For each above defined admissible sum we have Tx,oU : ~ Tl,oT(o;~ai ~. Hence T~,ou is represented by the corresponding admissible sum as element of F'[V~, W1]. It is now obvious that l] Tx,oU I!F'[r~, w~] ~ I] u [l~'[go, w0], which proves that /~' is a normalized interpolation method. Consider now the case of /~". Suppose u~F"[Vo, Wo]. By (14.2"), (13.6) and Theorem [13.VII, b')] we obtain that for every [Vi, Wi] E g, and every T~,oE'~,~ with tT~,ol~,o<~l,T~,ouEA~and liui{~,,[Vo,%l:su p sup I]T~,oI!~.. i t Ti, o li.o_<~ Consider then T~,oU for some T~,oE ~,o with [T~,o[i,o~___~ 1. For any T~,~ E ~,~ withiTi,~]~,~l , T#,~T~,ouEA,(since T~,~T~,oE~i,o)andsup sup IIT~,~T~,ouHAi~ i I Ti, 1 ll,i ~i sup sup [1 Ti, ou [l,~ -- ]I u tim, Iv., %]. It follows that F" is also a normalized i ]iTi, o ll, o~_~ interpolation method. [14.¥II] COROLLARY - Let [17o, Wo] be a compatible Banach couple and Ao an interpolation space between Vo and Wo with interpolation constant Yo,o -- 1. For any class of couples ~ containing IVo, Wo], there exist normalized interpolation methods F such .that 2'[Vo,Wo]- Ao. Among all these methods there are two e~ctreme, F' and F" so that for any other such method F, and any couple [V~, W~]Ee~ we have F'[V~, Wi]~[V~, W~]~tP"[Vi, W~]. The methods F' and F" are given by (i4.4) F'[V~, W~] -- B°(Ao) F"~v W,] -- CO(Ao) The proof follows immediately from the preceding theorem when e~ is replaced by ~, and J~ is replaced by the class composed of the single couple [Vo, Wo]. REMilCK (added in the proofs) - The following statement can be proved by using the axiom of choice and the well-ordering of all BA~ACH couples: if F is a general interpolation method, then for each Banach couple IV, WJ we can choose for F[V, W] a norm (equivalent to the given one) so that the method becomes normalized. N. ARONSZAJN - E. (~AGLIARDO: Interpolation spaces, etc. 117 BIBLIOGRAPHY [ 1 ] N- ARONSZASN, Associated spaces interpolation theorems and the regularity of solutio~ls of dif[erential problems. Proc. of Symposia in Pure Matematies, Vol IV, 1961. AMS. [ 2 ] n. P. CALDERON, Intermediate spaces and interpolation. Reports of the Conference on Functional Analysis, Warsaw, 1960. 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