By Guido I. Zapata
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Additional info for Functional Analysis, Holomorphy and Approximation Theory II
Sample text
THEOREM 12. For arbitrary Banach spaces E and F , = {f E H(E;F) : f ( K ) is relatively compact in weakly compact subsets K in El, E = L1), f E Hwsc(E) for all with the topology of uniform convergence on weakly compact subsets of (or if F (Hd(E) ,T Od) E F Therefore, if E $ L1 E. if and only if df E Hd(E) e E’. Finally, we mention some related open problems in connection with two special Banach spaces. ive James space J First, consider the quasi-reflex- 1151, which is a non-reflexive Banach space with basis which is isomorphic to its second dual.
Since f(K) E’ for e a c h u n i f o r m l y on e a c h s u c h dPn(K) s contained and i s t h e r e f o r e r e l a t i v e l y Pn is F i n a l l y t h e u n i f o r m convergence of t h e gives the r e s u l t . D. We remark t h a t we have no example of a Banach s p a c e which t h e lemma f a i l s . K. Theref ore n. D. PROOF O F T H E O F E M . f = CPn We F i n a l l y , we may w r i t e Thus by Lemma 2 . 4 as 0 dP(K) i s complete. df(K) (yj). one c o o r d i n a t e t e n d s weakly t o d i n a t e s a r e weakly Cauchy.
E = 41 I n any WEAKLY SEQUENTIALLY CONTINUOUS ANALYTIC FUNCTIONS 35 case, the above proof shows the following. COROLLARY 11. If K df(K) in E, f E H(E) Let is relatively compact in E Note that if E‘ f o r each weakly compact set for all is such that for every weakly compact subset is compact in [4]), and it is unknown if In other words, using [a], E. 3, Hwsc(E) = Hw(E) df(B) is an arbitrary Banach space. f E Hwsc(E). then Hw(E) = HwU(E) E where K c E df(K) it is un- is compact in E’ it necessarily follows that E’ for every bounded subset B C E.
