Journal Article10.4153/CJM-1995-035-1
Weak-Star Continuous Analytic Functions
69
TL;DR: In this paper, the authors consider the algebra of analytic functions on a complex Banach space with open unit ball B that are weakly continuous and are uniformly continuous with respect to the norm, and they show that these are precisely the functions on B that extend to be weak-star continuous on the closed unit ball of B.
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Abstract: Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.
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References
A uniform algebra of analytic functions on a Banach space
TL;DR: In this article, it was shown that any polynomial in elements of 3 * can be approximated pointwise on B by functions in A(B) of the same norm.