Complemented Subspaces of Sums and Products of Banach Spaces (*)

Complemented Subspaces of Sums and Products of Banach Spaces (*).

G. ~r - V. B. ~r

Summary. - We prove that, when X is one o/ the Banaeh spaces 1~ (l

O. - Introduction.

In the structure theory of Frdchet spaces, it is a general problem to determine what complemented subspaces of a given Frdchet space look like. So far~ the attention has been focused on the classical nuclear spaces of power series type and on some Dragilev spaces, for which a number of results are available (see [3], [4], [11], [13] and [14]). In particular, we mention that the situation is completely clear in the case of stable i nuclear power series spaces of finite type: thanks to results of B. S. MITJAGI~ [15], [16] (see also D. VoG~ [11]), complemented subspaces of such spaces are of the same type. However, all the above spaces have a continuous norm. For Fr~chet spaces without a continuous norm there are no results except the well-known fact that ~o always appears as a complemented subspace. It is the purpose of this paper to initiate the study of complemented subspaces in Frdchet spaces without a continuous norm by looking at some very special spaces of this kind. The spaces considered are of the form X ~v, where X is one of the familiar Banach spaces l ~ (l

(*) Entrain in Redazione il 25 febbraio 1987. The second author acknowledges partial support from the Italian Ministero della Pub- blica Istruzione. Indirizzo degli AA. : G. M~TAsu~: Facolt~ di Scienze M.F.N., Universits della Basili- cata, Via N. Sauro 85 (Rione Francioso), 85100 Potenz~, Italia; V.B. MOSCA~ELLI: Dipar- timento di Matematica, Universi~k, C.P. 193, 73100 Lecce, Italia. 176 G. ~r - V. B. ~OSCATELLI: Complemented subspaees o] sums, etc.

Finally, we conclude with an Appendix containing results of an independent interest and remarks on injeetive spaces. We use standard terminology (see, e.g, [5]) and, in particular, we denote by ~o (resp. by ~) the product (resp. direct sum) of countably many copies of the scalar field (~-R or C). Also, for two locally convex spaces E and ~, we write E ~ F and E ~/~ to mean respectively that E is topologically isomorphic to /z or to a complemented subspaee of F. We use also the by-now accepted term quo~ection (coined in [1]) for the projective limit of a projective sequence (E~, R~) of Banach sp~ces E. and surjective linking m~ps R~: ~.+~-+E~ and for this we introduce the notation quoj~ (E~, R~). l~on-trivial (or twisted) quojections, i.e. quojections that are not isomorphic to products of Banach spaces, were first constructed in [7] by the second author. A distinctive feature of all quojections, which will be useful to us and that can be desumed from Proposition 3 of [lJ, is that their quotients are again quojections and hence ]~anach if they have a continuous norm. :~inally~ we recall that a locally convex space E is primary if, whenever E=~G, then either F~ or G~E.

1. - Complemented subspaees of (c0)zr and (l~) N (1 ~p ~ c~).

Let E = proj, (E~ R.) be the projective limit of a projective sequence of locally convex spaces ~. and linking maps R.: E~+~-+ E.. By definition

"~ : { (xn)~ l~ E~t: Xn: "~Xn-~l for all n} is a (dosed) subspace of ~[ ~. It is clear that, for every strictly increasing sequence (n(k)) oN, we also have that E = projk (E~k),/~(~)), where the maps /~(~): E~(~+~)-+ 2~.(7~) are obtained from the R~'s by composition. l~ow, let P,~:/~-+E~ be the map defined by Pax ~-x~. Then R~P~+~-~ P~ and the following lemm~ holds:

~E~A 1.1. - _Let ~ be a closed subspaee o] E. I] ~.-~ _P~,(E) and 1~ is the restriction o] R~ to F~+~, then fF = proj~ (F~,/~.).

P~OOF. - Set G = proj. (/~,,,/~.). If x ~/~, then x = (x.) with x~= R~x~+l~tz~ for all ~. Hence x~ =/~.x~+l and x ~ (7. Conversely, if y ~ G, then y = (y~) with y~E_~, and y.:~.y.+l:t~.y.+~. For each n there is an x~eF such that _P~x~: y~. If k~ n, put R~: R~ ... R._IR~_I; then Pkx~: R.7oP~x': R.7r y~. This shows that limn x ~-- y and hence y e ~, since /~ is closed. Thus F ~ G alge- braicully and, as easily seen~ also topologically. G. IV~TAYUNE - V. :B. ~r Complemented subspaces o/ sums, ere. 177

TItEORElVs 1.2. -- Let X be one o/the Bancwh space l~ (l

PaooF. - We begin by representing X x as quoj~ (X '~, S,,), where S~: X~+I-> X ~ is the canonical projection (xl, ..., x+~, x~+~) --> (xl, ..., x~). Because E c X x, by Lem- ma 1.1 for each n there exists E~ c X ~ such that E ---- projn (/~, ~). Note that the maps ~: E,+~ :->/~, restrictions of the S,;s, are surjective, but the subspaces E~ need not be closed. However, since E is complete we may also write E----proj: (F~, S,,), where we have denoted by F~ the closure of/~ in X ~ and again by S, the restriction of S~ to F~+~, in general not surjective but with dense image in F~. Recalling that X ~ X ~, we see that each F. is isomorphic to a closed subspaee of X. Now we use the fact that E is a quojection to write E = quoj~ (E~, R:). If dim (ker R:) < co for all but a finite number of n, one easily obtains the first three cases. Therefore, by passing to subsequences if necessary, we assume s now on that

(1) dim (ker R~) : dim (ker S~) ----co for all n, and also that we have the situation indicated in the following commutative diagram

)- E~+2 :~"+~) E~+~ ~ > E. > ... (2)

... --+. F~+~ ~ F= ~ ..., where the maps R~ and L~ are surjecti+-e, while the S~ and Q~ need not be so. Introducing the canonical maps (see just before Lemma 1.1)

(3) P~: E -~E~ and P~: E -~Fn, one immediately has that

(4) P.---- L~P~ and P~----Q~P~+~.

By assumption there is a continuous projection T: XX-+E and we naturally introduce the maps

(5) T~= P~T: XN-* E~ and T~= P.T: XN-> F~ , for which we have 178 G..-YfE~A~'U~E- V. B. i~[0SCATELZI: Complemented subspaees o/ sums, etc.

I~ote that each 2"~ is surjective because so are P and P~ the Ia~ter in view of the fact that E ---- quoj. (E~, R,). Moreover, since E~ is Banaeh and Y~ is a quotient map, for every n there exists k(n)~ N such that T~, hence also P~_~, factors canonically through X ~(~~ i.e. the diagrams

X ~ ~ > ~ X N -----+~ j~

Xk(n) Xk(n+l) commut% where we have denoted again by T~ (reap. P~) the restriction of T. to X ~(~) (resp. of P~ to X~(~+~)). For the rest of the proof we consider three cases separately.

(a) X= ~" (~

Assuming, of course, dim E~= c~ we see that the map/'1: X ~(~) -->E~ is not compact, because it is surjeetive. Hence, by (6) the map T~: X~(~)->~F~ is not compact. Because F~ c X by construction and X ~(~) ~_ X, by [6] (Proposition 2.a.2 and the remark following Proposition 2.e.3) there exist sub@aces M~ c X ~(~) and ]/1 c/~ such that //1 ~_ X, H1 is complemented in X and the map P~: M~-~//1 is an isomorphism onto. Since S~P~+~= T,,, it is dear that also all the maps

are isomorphisms onto. l~ow note that Sl(x, y)= x for all (% y)~ F2 and hence, if N-~ ker $I, then 2V = F~(~ ({0}• Because S~: T~(M1) -+T~(M1) = H~ is an isomorphism onto, there exists a continuous linear re@A: M~--> {0} • with T~(M~) = {(x, Ax): x~H~}. It follows that, if i%: X-->H~ is a continuous projection, then the map r: X • defined by r(x, y) -~ (p~x, Aplx) is a projection onto T~(M~) with kerr = {@, y): plx = 0} = (kerp~) • Now, recalling the second diagram in (7), we consider the map P~: X ~(81 --> iv2. Since N ---- ker S~ r i~ and dim N ----- ~z, the m@P~:Pj~(~)-+2vc{o}• is not compact and again, by the argument above which also @phes to sub@aces of l ~, we can find M~ c X k(8) and H~ r N such that H~X, H~ is complemented in {0}• and the m@T~: M2->H~ is an isomorphism onto. As before, all the maps S.: P~+I(M2) -+P~(M~) are also isomorphisms onto. If I is the identity of X • X and q: {0} • X -+ H2 is a continuous projection, consider the diagram

XxX • s" >{O}xX ~ .~. G. 5/IETAFU~E - V. B. 5~OSOATELLI: Complemented subspaces o] sums, eto. 179

It is immediate to see that the composition map

s = qS~(I-- r): X• ~H2 is a continuous projection and that rs ~ O, sr = 0. From this, it easily follows that the sum T2(M~)~H2 is closed in F2, hence equal to T2(M~)Q~(M2)~--XOX and that the mapp~= r-~ s is a continuous projection of XOX, hence of F~., onto T2(M~) Q T~(M2) such that S~p2 = p~S~. Proceeding in this way, we inductively obtain for every n closed subspaces of F~ q~ of the form G~ = (~ T~(Mk) and isomorphic to X ~, as well as continuous projec- k=l tions p~: X ~ -> G~ satisfying

(8) S~p~+I---- p~S~ , so that S~(G.+I)= G~. If we now form the projective limit G of the spaces G. with respect to the restricted maps S.: G~+I -+ G~, we see that G c E and G ~_ X N. Moreover, by (8) the map p defined by the sequence (p~) is a continuous projection of X •, hence of E, onto G. We have the situation X N < E < X N and an application of Pelczyfiski's decomposition method in our setting (cf. [12]) completes the proof in this ease.

(b) X = co.

This ease is substantially simpler than the previous one. Indeed, it suffices to represent E as quoj~ (E.,R.) and to note that each 3Y., being a quotient of (Co)~(") ~ co, is also isomorphic to a subspaee of Co by [6], Theorem 2.f.6. Then, with the notation as in ease (a), we car~ take F. = ~. and proceed as before, but using directly the maps T.. A further simplification may be derived from the fact that once it is proved that E contains a subspace G _~ (Co)N, then G is automatically complemented in E by Corollary 3.22 of the Appendix.

In order to prove this ease, we need the following lemma which will be useful also later on.

LEM:~A 1.3. -- Let (X,) be a sequence o] Banaeh spaces and let E c ~ X~. I] E is a quojeetion, then also E (~ ~I X~ is a quojection ]or each k.

n§ PlC00F. - Let S~: ~-[ X~-+ X~ be the canonical projections, so that ]~ X~:

= quoj,~ X~, S~ . With the notation introduced at the beginning of the proof of the theorem, we may write E = proj~ (E~, S~)= proj~ (F~, S~). If P~ and P~ 180 G. 3~_ETA~UI~E - V. ]3. ~0SCATELLI: Complemented subspaees o] sums, etc. are as in (3), with reference to diagram (2) and to (4) we have

1-[ x. = {(x:) + ..... = 0} = - -1(0)=

Now put H~ : Q[*(0) and Hk+~ = R~+~(HT~+t-1),-~ so that H~ c E~+I and Hk+~ c E~+~+~. Then

(9) ~ ~ l-[ X~ : quoj~ (H~+._,, R~+,) n>~C and the lemm~ is proved.

I~EhL~RK !.4. -- Note that if E (~ ~[ X~ = {0}, then 15 is one-to-one, hence so are all the maps S~ and R~ for q'~>k and E is Banach. On the other hand, since Hk+ ~ = Rf11(H~)~ R[)~(0), we see from (9) that, under our standing hypothesis (1), E (h ~ X~ is va {0} and has the Banach space Hk+~ as an infinite-dimensional quo- n>k tient, so that /~ n !-[ X~ r o~. n>/c Now we are ready to finish the proof of the theorem.

PI~OOF OF CASE (e). - Again, we have to consider two cases.

(~) E is reflexive.

I~ this case all the Banaeh spaces F,~ are reflexive, as well-known, hence the maps T~: X k(') -+ 1~ (cf. (5) and (7)) are weakly compact. Because X k(~) _~ 1~ and 1~ has the Dunford-Pettis property, the maps Y. are also completely continuous. ~ow let B be a subset of E such that P~(B)= B~ = the unit ball of E~, for each n. B is bounded in E and hence weakly compact, since E is reflexive. But then P~T(B) : T~(B)= T~[%~)(B) is compact in E~ for each n and hence T(B) is compact in E. Since B ---- /~(B) because T is a projection onto E, B is compact and so must be each BE. = P(B). It follows that dim ]~ < c~ for ~11 n, which contradicts our assumption (1). Thus ~< X N cannot be reflexive unless E_~o.

(fl) .E is not reflexive.

In this case we may assume that none of the E~ is reflexive. Then the map T~: X *=(11 -~EI~ being surjective, cannot be weakly compact and, therefore, by Rosenthal's theorem (cf. [6], Proposition 2.fA), T1 is an isomorphism on a subspace M1 of X k<~), with M1 ___ l% As a consequence, for e~ch n the map

R~: :T~+I(M1) --> T~(M1)

is an isomorphism and ir.(M~)_~l ~. Now, by Lemma 1.3 and Remark 1.4, G. META~U.~ - V. B. hi0SCAT~.I.LI: Complemented subspaces of sums, etc. 181

E (-~ ~] X~ (X~ = l ~) is a qnojeetion ~ co. Denote by T the restriction of T to n>k(1) 1-[ X.. n>/c(1) Suppose that all maps _P.~ are weakly compact; then the same holds for their restrictions to E n ]-I X~. But on this space T is the identity and this implies the reflexivity of all the spaces H~+z-1 in the representation (9). Then, as in case (~), e~ch set P,~(B) ~ P~'(B), with B c E (~ [I X~, is compact in E~, because B is n>k0) weakly compact and the _P~T are completely continuous. It follows that B is compact in 1~ and hence in E c~ ~ X~. But then B~0(~I+~-~ is compact in Hk(~l+~-~ n>k(1) for each I and this implies E n ]-[ X. ~_ co, contradicting Remurk 1A. n>/c(1) Thus, we may assume that P2~: I] X.-->/~2 is not weakly compact and, as n>k(1) before, there must exist a subspace M2 c ~X~ such that M2 ~ 1~ and P~T: M~-+ n>k(1) ---~P~'(M~) c]~ is an isomorphism. ~7ow (P~T)]~ = T~, hence T~: M~-+ T~(M~) is an isomorphism onto and again R~,: T~+~(M~) --> T~(M~) is an isomorphism onto for n > 1, while R~T~(M~) : TI(M~) ~-- {0}. Iterating this procedure, as in case (a) we obtain spaces G~----@T~(M~) so that if G=proj~(G,,R~), then G c]~ and

G ~ (l~) x. Either directly or using the fact that (t~) N is injective (see the Appendix), it can be seen that G < E and again Petezyflski's decomposition method yields the desired result. The proof of the theorem is now complete.

COl~OlJ.AI~Y 1.5. - I] X ~ l~ (l

2. - Complemented subspaces of (Co)(N) and (/~)(N) (l•p< ~).

If X is a Banach space, we denote by X (N) the locally convex direct sum of countably many copies of X. The following result is the analogue (and only in part a consequence) of Theorem 1.2 ill the context of direct sums.

Tm~ORE~ 2.1. - Let X be one o] the Banach spaces I~ (1

Pl~ooF. - Since E is complemented in X (N), E is an (LB)-space and hence we may represent it as the strict inductive limit of the B~nach subspaces E~----E c~ X ~. If dim (E~/E~_I)< c~ (J~o---- {0}) for all but a finite number of n, one obtains the first three cases. Therefore, by passing to a subsequence if necessary, we may assume that

(10) dim (E~/.E~_~)= oo for all n. 182 G. 5J[ETAFUNiE - V. B. )/[OSCATELLI: Complemented subspaees of sums, etc.

This time we have to examine fore" cases.

(a) X=l~ (~

Since E< X (~), for the strong dual EZif we have EZ/ < (X') s, hence E~_/ (X') ~ by Theorem 1.2 and finally E ~ X (~) by reflexivity.

(b) X = l<

If E < (l!) (zv), then Be < (l~) x and hence Er ~ (l~) N by Theorem 1.2. But then (l~)(~)< ~ by Proposition 3.3 of the Appendix and again the theorem follows from an application of Petezyfiski's decomposition method in the setting of direct sums (cf. [12]).

(e) X = Co.

Observe that, for each n, E.c (co)~ ~_ co is a subspace of Co and E,~/E,_~ is a subspace of (Co)'*/E,_~. Since (eo)'~/E,~_~ is a quotient of Co, it is isomorphic to a subspace of co by Theorem 2.f.6 of [6]. Thus, also E~/E~_~ is isomorphic to a subspace of Co. Now tel 2~ _~ co be complemented in E~ (F~ exists by (10)). if q: E2 -+ E2/E~ is the quotient map, by (10) q is a non-compact map between two subspaees of So and hence there exists a subspace Y~ c E2 such that s 2'~(3 E~= {0} and F2 ~-E~ is closed in E~. Clearly _/~ d-/~ must be closed in E~ and isomorphic to Co@ co. Let p~: E~-+E~ be a continuous projection. If f~ is the identity of _F2, consider the m~p p~ ~- I~: E~ @ 2~ -+/~ @ _~ ~_ c0 stud extend it to a map p~: E~ --> -+ F~ @ ~ (this is possfble because /~ is separable and co is injeetive for separable spaces). It is easily seen that p~ is a projection such that Psls~ = P~. Continuing in this way, we construct a subspaee F of E which is isomorphic to (so)(~) ~nd complemented in N and the result then follows as in case (b).

Set X~--1 ~ for all n and represen~ E as ind~E~ with E,~=E(3 @Xi. If i=1 T: @ X, -~ E is our projection and T1 is its restriction to X1, then the map T~: X1 -+ -+Ek(~) is bounded for some k(1)s N. Suppose that there is an re>k(1) for which E,,~ is reflexive; then T~, considered as a map from X~ into E~,, is weakly compact and hence completely continuous. Because its restriction S to E~ c X~ is the identity I of E~, we get that I -- S 2 is compac% i.e. that dim E~ < c~. Since this contradicts oar assumption (10)7 we may suppose that none of the Banach spaces E,, is reflexive, which means that E is also not reflexive. But then the above map T~ is not weakly compact, for it is the identity on the non-reflexive Banach space E~; hence there is a subspgce -~[~ ~ 1| of X~ on which T~ is an isomorphism onto a subsp~ce 2g~cET~(~) and we denote by pl a continuous projection of ET~(~) onto 2V~ _~ 1% G. ~_ETAFUNE - g. B. ~V[OSCATELLI: Complemented subspaees oJ sums, ere. 183

~(1) § 1 k(1)§ If T~ is the restriction of T to @ X~, then the map T~: | Xi --> E~(~) is i=1 i=1 bounded for some k(2) a ~Y, k(2) > k(1). Consider the diagram

~(~)+ ~ (1]_) | X~ ~" where q is the quotient map. If qT2 were weakly compact, because T2 is the identity on E~(,)+~ the space E~(~)+,/s would be reflexive We show that this is not possible. ! Because E< | we have ~'~ < ~X: (X: = ~' for an ~) and ~ = ~ gt ! .! ./ ! l ----- quo L (E~, 1.), where the maps L: ~n§ are the duals of the canonical em- beddings j~: E.--> E.+~. We shall prove that no quotient space En+,/E~ is reflexive by showing that no dual (E~+~/E~)'= kcrj: is reflexive. ! ! ! To begin with, Ez is not reflexive, because E is not reflexive. Let S: ]-[ X~ -+ E~

! f be a continuous projection and put &= P~S, where Pk: E~-+E~ is the canonical surjection for each k. Because dim (kerj:)>dim (E~+~/E~)= oo by (10), we have that E'rh ~ X.' is a quojection ~ co by Lemma 1.3. Then E'rh ]-[ X'. n>~k n~Ir cannot be reflexive. In fact, as in the case of 1~, (l~) ' has the Dunford-Pettisproperty ((l=)'_~ LI(#)) and if E'n ~[ X: were reflexive, proceeding as in the proof of case (~) in Theorem 1.2 we would obtain that E!(~ ~ ~ ,-" co, hence a contradie- n~k tion. Now consider the maps &: I] X'~ -~ E:. By passing to subsequences if necessary, we may assume that S. X~ ----0 and also that S., regarded as a map from "i~n + 1 1-[ X'~ to E~', is not weakly compact. But then, considering the commutative diagram

E:,

it follows from &=0 on 1~ X~ that Sn+l ' cker " and we conclude that ker j; is not reflexive. ~>~+l "i>~%1 Going back to our diagram (11), we have established that the map qT, is not ko)+l weakly compact and hence there exists a subspace M~ 1| of | X~ on which ~=1 qT~ is an isomorphism onto a subspace of E~(2)/Ek(1). Putting N2 = T2(M~), we then have that N~ c Ek(~), N~ ~_ 1~, 5r~ Ch g0(~) = {0} and N2 q- N~(~ F @ l ~) is closed in 184 G. --~ETAFU:N:E - V. ~B. ~_[0SCATELLI: Complemented subspaces o/ sums~ etc.

E~(~). Continuing as in case (e), we find that E contains a complemented subspace isomorphic to (I~) (N) and the usual argument yields the result. This completes the proof of the theorem.

C0~0LLA~Y 2.2. -- I/ X~ l" (l

3. - Appendix.

In this final section we collect some tools that we already used and which may be of an independent interest, as well as some consequences of Theorems 1.2 and 2.1. First of all, we make some remarks. (All the spaces considered in this section ~re assumed to be infinite-dimensionaL)

~EMA~K 3.1. - Using the fact that a qnotient of ~ quojection is again ~ quojeetion, it is immediately seen that every infinite-dimensional quotient of (l~)~ is isomorphic to ~o, 1~, ~o • 1~ or (l~)~, while (l~)~ has alt nuclear Fr6chet spaces as subspaces (not complemented, of course). It follows, by duality, that every infinite- dimensional subspace of (I~)(N), which is also an (LB)-space, is isomorphic to ~, 1s, ~l ~ or (ts) (~), while (12)(s) has all nuclear (D/~)-spaces as quotients (again, not complemented).

RE)~A~X 3.2. - The proof of case (b) of Theorem 1.2 also shows tha~ for every infinite-dimensional quotient F of (Co)N, either F ~ co, Y~ w • ]( (Y a quotient of co), or /~ contains a complemented subspace isomorphic to (co)N. Similarly, the proof of case (c) of Theorem 2.t shows that for every infinite-dimensional subspace E of (Co)(N), which is an (LB)-space, either E ~ ~, Z, ? | Z (Z u subspuce of Co), or E contains a complemented snbspgce isomorphic to (Co)(N). We now give a proposition which is the essence of the proof of case (b) in Theorem 2.1 and which has a number of corollaries.

! P~OPOSITIO_~ 3.3. - Let E be a strict (LB)-spaee. I/ E~ has a subspace isomorphic to (co)N, then E contains a complemented copy o/ (/1)(N) and, consequently, (l~)N is a subspace o/ E~./

y P~ooF,- Let T: (co)x-+E'~ be an isomorphism into, so that T': Ep __~(/1)(N) is a quotient map. Put X~= 11 for all n, so that (ll)(N)= O X~ and let

P~: (~)X~-+ X. be the n-th canonical projection. For each n~ P~ is a[(ll) (N), (co)N] -- i -- a(l 1, co) continuous and hence P~T~: EH-~X. is a(E ~, E') -- a(/~, co) continuous. Write E -~ ind~ E~, where the increasing sequence (E~) of Banach spaces is strict. g Let B~ be the unit ball of X~; there is a bounded subset C of/i)~ such that P~ T'(C) ~ B~. Then, if k(1)e N is such that C c E/I(~ the open mapping theorem implies that the map P~T': E~(1)-+X~ is surjective and open. As in [6]~ Proposition 2.e.8~ there is G. META~U~E - V. B. ~VI0SCATv,LZ~: Complemented subspaces o] sums, etc. 185 a subspace G~ of X~ and a continuous projection rl: X~->G~ such that the map L~:/~(~) -> G~ is surjeetive, where L~ ~- r~P~T'[~. Denote by (e~) the sequence of canonical vectors of Gx ~ ll; there is a bounded sequence (x~) in Ek(~) such that Z~x~ = e ~ for all j. Now extend the map e ~ --~ x ~ by linearity and continuity to map s~: G~-~E~(~. Then s~Z~ is a continuous projection of E~(~) onto s~(G~)~ G~. Because E~(~) is Banach, there is an h(2) e N such that P~T'I~;(.)~- O for all n~>h(2). g Then, as before, we can find a k(2) e N for which the map P~(~)T': E~(~)---~X~(~I is surjective and, moreover, there exists a continuous projection r~: X~(21-~ G~ c X~r G~ ~ 1~, such that the map Z~: E~(~I->G~ is surjective, where L~ ~ r~P~I~IT'[~. :Note that L~]~)= O. ~ow we show that the mapL~-L~: E~(~)-~ G~O G~ is surjective. In fact, if x ~ G~ there is a z ~ E~(~) such that x ~ Z~z = (L~ + L~)z. If, instead, x ~ G~, we can find a z e E~(~) for which x--~ Z~z ~ (Zt~-Z~)z--Llz. Because L~z ~ G~, there exists y e/~(~) such that L~z ~ Z~y -~ (Lz ~ L~)y and hence x ~ (Z~ -~ L~)(z-- y). Thus L~ ~ L~ is surjective. :Now denoting by (e~) the sequence of canonical vectors in G~ ~ I x, we can find a bounded sequence (x~) in ~(~) satisfying (L1-~ L~)x~ ~ e~ for all ~. Again we consider the map e~-~ x~ and extend it by linearity and continuity to a map s~: G~-~ J~(~). Then the map (L~-~ L~)(s~-s~) is the identity on G~G G~ and hence (s~-s~)(L~-]-L~) is a continuous projection of E~(~) onto (s~ + s~)(GI~ G~) ~ s~(G~) ~- s~(G~) ~ G~ G~ and, clearly, s~(G~) -~ s~(G~) = s~(G~)~ ~)s~(G~). Moreover, (s~-s~)(L~-L~)[~(.)--~ s~/~ since L~= O on E~(~) and s~----0 ou G~. Iterating this procedure, for each n we find a k(n)~ N and a continuous projection p~ of E~(~) onto a copy of G~...~G~ satisfying P~I~(,-~)-~--P~-I. To complete the proof, it is enough to notice that the map p : E --> (~) G, ~ (P)(~), defined by the sequence (p~), is a continuous projection.

CORO]~LARu 3.4. -- Let ~ be a strict (ZB)-space. I] (p)(N) is a quotient o] E, then it is also complemented in ~. Moreover, this property characterizes (p)(N) among all separable and strict (LB)-spaees ~ % P or ~Q 1~.

PROOF. -- Let Q:/~ -> (l~)(N) be a quotient map. Since E is a (DE)-space, the dual mapQ': (1)~~ N ~-~ E~, is an isomorphism and hence, by Proposition 3.3, (p)(N) < E. This proves the first assertion. Yor the second one, let F be a separable and strict (LB)-space c~ ~, 11 or ~G t 1 and with the property that Z < E whenever E is a quotient of/~, with E an arbitrary, strict (ZB)-space. Writing F : ind~ F., where the F~ are Banach spaces, from the separability of F~ we obtain that F~ is a quotient of 1~ for each n. It ~ollows that F is a quotient of (p)(N) and hence that E < (l~)(N) by the assumption on T. But then F ___ (l~)(N) by Theorem 2.1.

COROLLARY 3.5. - (Co)~ is contained in no separable dual o] a strict (ZB)-space.

COROLLARu 3.6. -- ~or any open subset [2 ~ r o] R ~, the space C([2) o] all continuous ]unctions on [2 is not isomorphic to the dual o] any strict (]SB)-space. 186 G. lV[ET.4~U~E - V. B. M0SCAT]~LLI: Complemented subspaces of sums, etc.

Now we observe that, using techniques similar to those employed in the proofs of Theorems 1.2 and 2.1, one can obtain the following results.

PnoPoslTIO~ 3.7. - For each n let X~ be a Banach space with the Dun]ord-Pettis property.

(a) I]E < ~ X~ and E is reflexive, then E ~ o~.

(b) If ~ < @X~ and E is reflexive, then E ~ ~.

P~ol~osI~IO~ 3.8. - For each n ~et K~ be a compact space. I] ~ < ~ C(K~) and

E ~ m, X or o • with X Banach and X< C(K~), then (Co)~CE (and, hence, ~1 i] .E is separable, (co)~< E by Corollary 3.22 below). If, moreover, the K~ are stonian, then (l~)~< E.

Here, instead of Rosenthal's Theorem on maps from 1~, one uses a we]l-known theorem of Peiczyfiski on maps h~om a space C(K) (cf. [8]).

R~A~K 3.9. - The second assertion in the above proposition continues to hold when Z < [C(flN~N)] ~ -- (t~/Co)~ even though fll~N is not stonian.

COROLLARY 3.10. -- Let ~ V= r be an open subset o] R~: I] E < C(~) and E ~ ~, X or o~xX~ with X Banach, then (co)N< E.

PRo0~. - It suffices to note that C(~)~ (C([0, i]))N (see [I0], Chapter 3, w 3.6) and to apply Proposition 3.8.

P~01)OSlTIO~ 3.11. - Let (#~) be a sequence o] (~-]inite measures. If E < @ L~(#~) and E ~ qJ, X or q~@X, with X Banach and X< @ L%u~0, then (I~)(N)< E. ~=1

P~OOF. - It follows from the assumption that Ezf < ~[ L~(#.), hence EZ/ ~ (Co)N % by Proposition 3.8 and the assertion is an immediate consequence of Proposition 3.3.

P~o~osI$IO~~ 3.12. - ~or each n let K~ be a stonian space and suppose that E < ~ C(K~). I] E is separable, then E ~_ (~. ~t

PROOF. -- Write E ~ quoj~ E~, where the E,~ are separable Banach spaces. For each n let P~: E -~E~ be the canonical quotient map as in (3). If P: y[ C(K~) ->E is a continuous projection, then P~P: ]-[ C(K~) ->E~ is a quotient map and hence k(n) there is a /~(n) ~ N such that E~ is also a quotient of ]-[ C(KJ ~_ C(K) for some i=1 U. ~_ETAFUNE - V. B. ~V~OSCATELLI: Complemented subspaees o] sums, etc. 187 stonian space K. Since E~ is separable, it must then be reflexive (this follows from the theorem of I~osenthal already quoted), and since this holds for every n, E itself must be reflexive. But then E ~ ~o by Proposition 3.7(a). Finally, we turn our attention to injective spaces. A locally convex space is injeetive if it is complemented in every locally convex space containing it. As in the case of Banach spaces (see [6], Proposition 2.f.2), one can show that (l~) J is injeetive for any index set J and that a locally convex space E is injeetive if, whenever F and G are locally convex spaces with F c G, then every continuous linear map T: F-+ E can be extended to a continuous linear map T: G--> E. We also observe that:

(a) if dj is a cardinal number for each j e J, then l-[ Id~ is injeetive;

(b) if Xj is an injective space for each j e J, then l-[ Xj is also injective; J~J (c) a complemented subspace of an injective space ia again injeetive.

The following lemma generalizes to locally convex spaces the stituation en- countered in the case of quojections (ef. [1], Proposition 3).

~LE:iVIMA 3.13. - Zet E be a complete locally convex space. E is the strict projective limit o] a ]amily (J~j, t~j: i, j ~ J) o] Banaeh spaces J~j and sur~eetive linking maps ~i~ : Ei ---NEg i] and only if every quotient o] 1~ with a continuous norm is Banach.

PgooF. - Let E = proj (Ej, _R,j), let F be a quotient of E under a quotient map Q and suppose that/~ has a continuous norm. It is easy to see that Q factors through some Banaeh space /~j, i.e. Q--QjP~, where Qj:Ej---~t z is continuous and P~: E -+/~. is the canonical quotient map as in (3). Clearly Q~. is surjective, because so in Q. But Q~ is also open, because P~. is continuous and Q is open. If follows that Q~ is a quotient map and hence F is Banach, as a quotient of Ej. Conversely, suppose that every quotient of E with a continuous norm is Banach. Let (pj: j e J) be a fundamental system of continuous seminorms on E. For each j e J consider the space /zj--_ E/kerp~ with the quotient topology. Because E is complete, E is the strict projective limit of the family (Fj: j e J). But every F~. has a continuous norm and hence is Banach by assumption.

P~oPosI~IO~r 3.14. - Let E be an injective locally convex space. Then one (and only one) o] the ]ollowing cases must occur: (a) E ~ c% ]or some cardinal number d; (b) .E ~_ X ]or some injective Banach space X; (e) /~--~m~xX; (a) Cl~)N< ~. 188 G. ~ETAFUNE - V. B. ~/J~0SCATELLI: Complemented subspaces el sums, etc.

PI~OOF. - E is certainly a subspuce of a product ~ E~ of Banach spaces and hence J~J also a subspace of a product of the form [I ld~ Because E is injective, it must be Jez complemented in ]-~ l ~176 Then every quotient of B with ~ continuous norm is also ieJ a quotient of ~ td~ and hence is Banach by Lemma 3.13. It follows, by the same J~J temma, that E is the strict projective limit of a fumily of Banach spaces. At this point, as for Proposition 3,8 the result can be achieved by using techniques similar to those employed in the proof of case (c) in Theorem 1.2.

COrOLLArY 3.15. - Every injeetive ~ocally convex space has a subspace isomorphic to o9 or 1~.

COI~OLLAI~u 3.16. - If E is a locally convex space which is injective and reflexive, then E ~ c% ]or some cardinal number d. Hence co is the only injective and separable ~rdchet space.

R~A~K 3.17. - In the second assertion of the above corollary the assumption of being Fr6chet is essential, since ~o~ (c ~- the cardinality of the continuum) is injeetive and separable. Indeed, ~o and coc are the only locally convex spaces with this property. We now give the following characterization for injective spaces of continuous linear maps.

PROPOSITION 3.18. - Let E, F be two locally convex spaces and denote by ~(E, F) the space o] all continuous linear maps/tom E into ~ endowed with the topology o/uni- ]orm convergence on the bounded subsets o] ~E. I] E is quasi-barrelled, then the ]ollowing assertions are effuivalent:

(i) ~(.E, F) is injective;

! (ii) EZ and F are injective.

! PROOF. - (i) -~z (ii): This is clear, because both Ep and F are isomorphic to complemented subspaces of ~a(E,/~). (ii) ~ (i) : Since F is injective, we have F < ~-~ ldjco for some index set J and car- dinai numbers dj, j ~ J. If follows that J~J

g~(E, ~') < g~ ~, Id~ = ~ g~(E, le, ) jeJ and hence, by what we have aire~dy observed, it suffices to prove the injectivity of a space of the form s l~) for an arbitrary cardinal number d. G. META~U~E - V. B. lV~OSC~E~.L~: Complemented subspaees o] sums, etc. 189

Since E is quasi-barrelled, we have that s 1a ) ----ld co (Ez) , by [2], Proposition 5.14. l~ow the injectivity of/~ r implies that 1~f < I] l~ for some index set H and cardinal numbers m~. It follows that h~

l~ (E:) < l~~ l~= = I] l:(l~=} ~heH ~ heH and the assertion follows upon noticing that the third space is injective, being isomorphic to a space of the form l~ lg7 for suitable cardinal numbers g~. hell

I~.~A~K 3.19. - The injectivity of g~(E, F) is already known and can be established directly by tensor product methods when the spaces Ezl and F are drawn from the familiar injective spaces cog and/or L~(#). E.g., the injectivity of g~[L~(#), Z~(v)] (#, ~ a-finite measures) is quickly established by the following chain of isomorphisms:

However, these methods do not work when E is an injective space which is not isomorphic to a dual space. Thus, a non-trivial case of an injective space ~(E,/~), which cannot be obtained by the above methods, follows from Proposition 3.18 by taking, e.g., for E a space JS~(#) and for F one of the spaces C(K) (K stonian), not isomorphic to dual spaces, constructed by l~osenthal in [9]. We conclude with a brief look at the separable case.

P~oPosITIo~ 3.20. - I] ~ is a separable locally convex space containing (Co)c (c : the cardinality o] the continuum) then (Co)~ is complemented in E.

PnooF. - To prove the assertion it is clearly enough to show that Co < E whenever Co c E and this can be achieved by a simple extension of the proof which is used in the Banach space case (see [6], Theorem 2.f.5).

COI~0LLA~Y 3.21. -- The spaces op, c%,Co, r215 o9~• (co)N, wo• N and (Co)~ are injective in the category o] separable locally convex spaces.

P~oo~. - The assertion follows from Proposition 3.20 and the fact that all the spaces in the statement of the corollary are complemented in (Co)~.

COROLT.A~Y 3.22. - The spaces co, Co, co • Co and (Co)N are injeetive within the class o] separable Frdehet spaces. 190 G. !~ET~UNE - V. B. ~OSCATELLI: Complemented subspaees o] sums, ele.

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