By João B. (Ed.) Prolla
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Additional resources for Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory
P. 6ote. each 2 2 U E Ui- • [ 1 M tead 06 th,t-1, we could al-1o ftequ,tfte HF (K ) 2 oft 2 1 HF (K 2 ) to have the. a. p. I . 2 HFI E F2 (K l xK ) 2 l Then v hofd-1. HF (K l ) 0 HF (K 2 ) E 2 1 0 (2) 16 Kl and K2 aILe "6a:t", J... e. t,[-16y K. =K. 1- 1- (i 1,2) , we get: PROOF: (1) The remark in brackets is obvious from 6. (3) and Schwartz's theorem. For the proof of (1) under the assumption on Fl resp. F , 2 50 BIERSTEDT So let set > 0 € U containing Kl x K2 be given and find and a function g E (F sup jf(x) - g(x) [ < xEK xK l 2 Without loss of generality we may assume l an sF )(U) 2 open suchthat s "2 U.
N Kor (:IC, £) . t. This corollary settles Example 1. It contains the corollaries of Proposition 3 as special cases. £) equals X We have seen that in the relative theory one cannot expect a similar result out an additional assumption on £. (X) of (in the sense of Choquet). £ leads to the complete generalization of Theorem 1. ai if the state space S (£) is a simplex. ve. ne: Kor The proof given in Lazar   (:IC, £) = "'£ 3C • makes use of the selection theorem of for (metrizable) simplexes.
P. 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 ).
Approximation Theory and Functional Analysis, Proceedings of the International Symposium on Approximation Theory by João B. (Ed.) Prolla