AUXILIARY CONCEPTS ASD FACTS IBTERNAL REPOET (.Limited distribution) 1. If {e } , {h. } are orttLonormal tases in the Hilbert space H, and n n 1 o> , respectively, and X > 0 are the numbers such that the series > X International Atomic Energy Agency n _ / . n
•,., and converges, then the formula Af = X (f,e )h defines an operator of the n n n Un'itad nations Educational Scientific and Cultural Organization n = 1 Hirbert-Schmidt type, and defines a nuclear operator vhich maps the space INTEBNATIOtTAL CENTRE FOR THEORETICAL PHYSICS H into if X < => n n = 1 By a nuclear space we mean a locally convex space with the property that every linear continuous map from this space into a Banach space is i [8,9,10] nuclear ' .
2. The inductive limit I ind. * (T ) is defined as follows: We are a« A a a CRITERION FOE THE IUCLEAEITY OF SPACES OF FUNCTIONS given a family {t (T )} of locally compact space, a linear space J and OF INFINITE NUMBER OF VARIABLES * a linear map u : * such that the family {u (4 )} spans the entire space * . The topology T , if $ is defined as the finest locally
I.H. Gali ** convex topology such that the maps A. : $ (T } •+ *(T),remains continuous. A International Centre for Theoretical Physics, Trieste, Italy. basis of neighbourhoods of zero may be formed by sets T A(U ), where U are neighbourhoods of zero in $ (T ) . If $(T) is a Hausdorff space, then it is called the inductive limit of the spaces 4 (T ), and the topology T is called the inductive topology. We may also write ABSTRACT
In this paper we shall formulate a new necessary and sufficient condition »(T) = H.ind (T. for *(T). see [8]. for the nuclearity of spaces of infinite number of variables, and we define new nuclear spaces which play an important role in the field of functional analysis To define the projective limit, let be a subspaee of © $, whose and quantum field theory. Also we shall give the condition for nuclearity of the i£ A 1 infinite weighted tensor product of nuclear spaces. elements * = { satisfy the relation 4. = u ( MRAMARE - TRIESTE u. . : * • *. < 1) . [8,9,10] August 1977 ID 3 [4] Wolka has given an interesting criterion far the nuclearity of inductive * To be submitted for publication. limit and projective limit of Hilbert spaces. ** On leave of absence from Mathematics Department, Faculty of Science, Let $ - I ind. H where H are Hilbert spaces, then it follows Al-Azhar University, Naser City, Cairo, Egypt. n ••+ « n " easily that $ is nuclear if and only if for every m there exists n > a. such {3,10,11] that the injection u : H •+• H is of the Hilbert-Schmidt type. nm m n 3. The infinite weighted tensor product of Hilbert spaces! The following Define the space $„ as the projective limit of the Hilbert spaces H in the 2 3 h 1Q construction of the space will be used further as shown in [5,6) . Let I > > ' ] {H.)._. be a sequence of Hilbert spaces H. , let e = Ce ).__ > Ce1€K.) be a fixed sequence of unit vectors of these spaces and let & = (i.)._, (S. > 0) be a fixed numerical sequence. The Hilbert space constructed below TeT will be called an infinite weighted tensor product Definition We shall say that the system of weights P has the property (H) if: i=l;e;6 1 00 x' T " for every T'fiT there exists T"ET: E P /P < » (l) of the spaces H. with the stabilizing sequence e and the weight 5 k=0 K k h. Recall that an operator C acting from a separable Hilbert space H and then ve can give the condition for the nuclearity of $ by; to a Hilbert space H is of the Hilbert-Schmidt (HS) type if for an orthonormal basis e ,e£,..., of the spaces H the series Theorem 1 The space $ is nuclear if and only if P has the property |C| = £ C e. Proof Let P be an arbitrary weight from P and H , be the corresponding converges. If H^ is contained in H,, , we say that the inclusion is quasi- Hilbert space. If P has the property H , then according to (l) there exists T" nuclear if the inclusion operator is of the Hilbert-Schmidt type. The norm a weight P €P such that |c| is called the Hilbert norm of the operator C .see [11. For the chain of the Hilbert space , see [1 ,12] • P /P T k'K kK k=0 In order to formulate our main problem we need to do the following: a) Let an orthonormal basis in a separable Hilbert space Let {(Pv ) ev'v ^e an orthonormal basis in the corresponding space Hj, HQ and let In T» for the weight P . If 0 , : H „ •* H , is the inclusion operator then T P = (P = (P.)™ : ...; PQ = 1; > 1 for every T" -1/2 °° T> T" k,TET} | (Pk | -r' k=0 +T' k=0 {2) be a system of weights. Then by every weight ve define the corresponding Hilhert space in the form: and,conversely,- from the last equality it can easily be shown that if * is nuclear, then P satisfies the property (s). 2 2 T r T= (u | = I |C|P V < -»(c)-,(c " EC™= + k=0 k k b) Now we consider H = H to bethe corresponding adjoint space of an antilinear functional on HQ , (e^l^.Q (be an orthonormal basis on it and *p to denote the adjoint of $„ . How 4' is defined as the inductive limit of r r , the Hirbert spaces K where H and * have the form: the spaces>[_T , vhere w. e , (w >" eC°| k k k k=0 and From the above we can prove the following. Lemma 1 *; = U H = {w = Z we. [wu),. _ec"l I |w < °» for TET -T k==0n k k k k=o (3) The adjoint spaces for K , Since tar the spaces l3T)TtT we find e^^VtsT , ||eo||+T = 1 and 1 for every T«=T. Then for every T = , where T = Tx Tx ... we can define the infinite tensor product 3"C = ® H. constructed ~by the stabilizing sequence e = (eQ,e ,...) as in [5,. 6 } . The proof of this lemma can easily be shown as in Eef, [1]( Chapt.l). Since Let J-Tj be the infinite tensor product of H constructed by the 111-II ill-IL . stabilizing sequence e = (e ,e_,...), and e = e ® e (§ ... be considered -p T as an orthonormal tasis in HQ , and e^ = {'e^g = (Fa ^-1/2 __ _ ,p v,-l/2 e 1 Oy a be an orthonormal basis on >T • We define the projective limit of the Hilbert spaces J-[ by the form At the end of this article we shall obtain the criterion for nuclearity of the infinite weighted tensor product of nuclear spaces. We have the following definition. Definition Then J-[ and t have tbe following form: The system of weights P satisfies the property N if for every T'eT , £ > 0 there exists T"EI such that < 1 + e . k=0 emd {u = j UaV Then we have the following theorem. Theorem £ The space nuclear If and only if P has the property N Since || eJ| =1 for every TeT we can define H T = & H for every Proof TET constructed by the stabilizing sequence e = (e ,e ,...) with basis 'e '); Let ", tfhi~h has the property S and :' - {l.)._ . be an arbitrary (P. Define the space as inductive limit for element, we can choose T" = {T.)._. such that -5- -6- a) The space z p i /p J- < i + — t i = 1,2,. Consider the Sobolev space W^CS ,dS(.t}) aa the completion of trigonometric k=0 k K - 2i iKT polynomials T (t) C, e ,n = 0,1,..., ty the norm The Hilbert norm for the inclusion operator O l ; H * H 1 L2(S ) T • T " 1 i i - EC, has the form 1 X*\ IT? T l T and define the locally convex linear topological space KCS ) = (I ^pCs ) |O i| = E P VP i ' m=0 fi k0 k k to k=0 k k with topology/be considered as the inductive topology of the Sobolev spaceJ[71 It is clear that K(B ) ia considered as a space of infinite differentiate functions on S Then ve have (o I < » , which gives the condition for nuelearity of 3> , vhere Ti p Definition + 0 , :J-f ii HTi is the inclusion operator. Consider the space K(T ) K(s ) to be reear^ei-as the Conversely, suppose that $_ is nuclear, and if T'£ T, e > 0, then we i = l-,e can find T" - (T.) such that the Hirbert norm of the inclusion operator infinite direct product of spaces constructed by the stabilizing sequence is convergent, i.e. } [71. On the other hand,let HQ = L2(S ), ek= e , k. = 0,1,2,..., and 111 , then 10 I ' -1 •• 0 as i P = E = {Pc , P? = (1 + |k|) } 1° k m=0 T! T1 I 1 i 2 and we can choose i- such that i > i~ , (o , ] - 1 •§ e , which is the condition be a set of countable weights. It is easy to see that P^ = 1 for every of nuclearity when i > i . m = 0,1,... and P^ > 1 for every m, k ,and £ satisfies the condition of nuclearity W . For every weight P™ we can write the corresponding space in the form: II. NUCLEAR SPACES OF FUNCTIONS OF-INFINITE NUMBER OF VARIABLES In this section we shall construct two illustrative examples for these spaces. K™ = {u(t> = E u.eikt| + [k|) <-> k=0 ' +m Let L (s\ dS) Tse a Hilbert space with complex-valued functions which are summable with respect to normed Lebesgue measure dS = — dt (tcR ) and (e. )_m 00 1 1 be its orthonormal basis. Let T = S x S x ... -T- -8- and the corresponding spaces for every index m = Cra-)°° can be written in the form: and ve can define A[S ) as the inductive limit of the spaces A^Y Definition i(ctnx, + ... + a A(T ) = x 1 1 * i=l;e defines • the infinite tensor product of spaces with weights Q7]). X ) i Theorem It Then we can define K(.T™5 as the projective limit of the spaces K^CT"), i.e The space c/)(T°°) is nuclear. This fact can easily be proved since the system of heights W 8atisfies the condition of nuclearity K . K(T ) = P, m= (m. ) , l 1=1 and we can prove that the space K[T ) coincides with K(T ) in the topological sense. Theorem 3 Using the above-mentioned it can easily be shown that the space K(T ) is nuclear, since the system of weights E satisfies the condition H . ACOOHLEDGMEHTS b) The spac The author wishes to thank Professor Abdus Salam, the International 1 Let HQ = L^S ), P = W be a system of weights such that Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. ) |p k=0 k m=l It ia clear that this system of weights satisfies the condition of nuclearity N and the corresponding space for each weight has the form Am(S1) = {u(t) = k=0 -9- -10- REFERENCES 1) Ju.M. Bereaanskii, Eigenfunction Expansion of Self-Adjoint Operators. Kiev 1965 (A.M.S. Providence, Rhode Island 1968). 2) A, Grothendieck, "Produits tensorials topologiques et spaces nuelears", Mem. Amer. Math. Sac. 16. (1955). 3) P. Kristenson, M. Mejibol and T. Poulson, "Tempered distributions in infinitely many dimensions - I: Canonical field operators",Math. Phys. 1, No.3 {1966). •4) I. Wolfca, "Reproduzuierende kerne und nuckleare Raume - I".Math.. Ann. 163,167-186 (1966). 5) Ju. M. Berezanskii and I.M. Call, "Positive definite functions of infinitely many variables in the layer", Ukrain. Math. Z. 2h_, h (1972). 6) Ju.M. Bereasnskii, I.M. Gali and V.A. Zuk, "On positive definite functions of infinitely many variables", Akad. Kauk S5SR T203, 1 (19T2)• 7) I.M. Gali and S. Zaud, "The infinite weighted tensor product of nuclear spaces" (in press). 8) K. Maurin, 'Methods of Hilbert space", PWN- Polish Scientific Pub. 1967. 9) A. Robertson, Topological Vector Spaces (CaF'irir'no Hniv. 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