E′ and its relation with vector-valued functions on E

Abstract

Abstract. We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on E and F, we show that the spaces of X-vahed n-homogeneous polynomials and analytic functions of bounded type on E and F are isomorphic whenever X is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on E and F are isomorphic without conditions on the involved spaces.

Extension of vector-valued integral polynomials

“…However, by Proposition 8 (and the analogous result for Pietsch-integral polynomials in [7]) AB(P ) is an integral polynomial if P is. It is natural to ask if AB(P ) admits an integral expression involving the measures that represent P .…”

“…Next proposition states the analogous isometric result for G-integral polynomials. Although it could be deduced from [7], we give a direct proof for the sake of completeness. Proposition 1.…”

“…In [7] it is shown that the Aron-Berner extension of a P-integral polynomial P : E → X is a P-integral polynomial from E to X, with the same integral norm. This statement involves two facts.…”

The Aron‐Berner extension, Goldstine's theorem and P‐continuity

“…On the preservation of the usual classes of polynomials through the Aron-Berner extension, it is clear that for finite type, approximable, integral, nuclear, extendible and regular polynomials are preserved, see [4] and [8]. But during the last few years it remained open whether the Aron-Berner extension of a P -continuous polynomial is P -continuous.…”

Isometries between spaces of homogeneous polynomials

“…In addition it is shown that this result is also true for the classes of nuclear, approximable, K-bounded, integral, extendible nhomogeneous polynomials along with the space of n-homogeneous polynomials which are weakly continuous on bounded sets irrespective of further conditions on E or F . In [13] these results are extended to spaces of vector-valued homogeneous polynomials although the techniques required are different. In [36] we are provided with a method of constructing an isometry of spaces of homogeneous polynomials on E and F from an isometry of E into F as follows: Given a Banach space E we use J E to denote the canonical embedding of E into its bidual E .…”