Canonical mappings for polynomials and holomorphic functions on Banach spaces (Q2723484)

The present article continues the research of several authors about functional representations for the canonical image into the bidual of spaces of holomorphic functions or polynomials on certain Banach spaces. If \(U\) is an open subset of a Banach space \(E\), \(H_\omega(U)\) denotes the subspace of \(H(U)\) of all the holomorphic functions which are locally (with respect to the norm) uniformly weakly continuous. If \(U\) is a balanced open subset of a separable Banach space with the bounded approximation property, then the space \(H_\omega(U)\) endowed with the Nachbin ported topology \(\tau_\omega\) is ultrabornological. Whe \(U\) coincides with the open unit ball \(B_r\) of radius \(0< r\leq\infty\) centered around \(0\) in \(E\), a Taylor series representation of the bidual \((H_\omega(B_r), \tau_\omega)''\) is obtained. A functional representation is derived when \(E''\) has also the Radon-Nikodým property and the approximation property. The Tsirelson-James space \(T_{J^*}\) satisfies all the hypothesis. In the interesting, long introduction on polynomials and holomorphic functions of bounded type, the author forgets the following two important contributions by \textit{M. Valdivia} which are not quoted in the paper [Proc. Am. Math. Soc. 123, No. 10, 3143-3150 (1995; Zbl 0848.46030) and Math. Nachr. 181, 277-287 (1996; Zbl 0860.46026)].