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Approximation Theory and Functional Analysis, Proceedings of by João B. (Ed.) Prolla

By João B. (Ed.) Prolla

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Space sheaf of E-val ued holomorphic functions. When for short, A(R,E), H(R,E) instead of E, OE 0 on the prodis just F = 0, we wi 11 the write, AF(K,E), HF(K,E), respectively. F = 0, some of the results in sections 1 and 2 are appar- ently part of the "folklore" of the subject, but usually not easily For accessible in the literature: We have already pOinted out in the intraduction that this paper is based on a generalization of the "old" article [2 1. Later on (in [1 ), section 1), o. B. p. After the proper change of notation and some identifications (using the fact that each Banach space is a closed subspace of C (K ') for some compact K') his resul ts there are quite similar to our theorem 3 (for Banach spaces E) .

AF (K,E) = HF (K,E) holds for all complete Lc. spaces by Corollary 5, too. In fact, Corollary 5 demonstrates equ~valent. the two approaches which we have just outlined are REMARK: Similarly, if then the a. p. c. space and i f AF(K) =HF(K), E or of AF (K) = HF (K) also implies AF(K,E) =HF(K,E) gene~al. 2. APPROXIMATION ON PRODUCT SETS Let us now turn to a description of the £-product plete £-tensor product of two (or more) spaces of type resp. com- AF(K) resp. HF(K). 6Uee p~oduct theo~em" for subspaces of, say, C (K l uct theorem was first stated in Eifler (17), (This slice prod- x K ).

K ) 1 l. l E ... , (i =l, ••• ,n) F (K n 1 x . . x K ). d. too. e.. c. complete space 10. COROLLARY: (2) (1) Let E E (see sect10n 1), we get e. g. : Let K , ... ,K 1 be nat. Then n be comptete and tet, , Fi (U ) i i (i =1, ••• ,n). 6 06 HF l E ... F (K e: n 1 oOlt. p. l. 6 x . . 6 valU. omplete, let K. ) i l. eaeh 1 =l, ••• ,n. 6 have the a. p. AF 1 e: ... e: (i =1, ... ,n) 1. = HF i 1. A . ) 1 Fi Fl. l. 6. too. (2). Let us remark that, e: ... p. for each Ul P : = {U 1 x • •• x Un of neighbourhoods of 9.

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