# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Weighted spaces of holomorphic functions on Banach spaces. (English) Zbl 0960.46025

This paper concerns functions on a balanced, open subset $𝒰$ of a Banach space. It builds on the work of R. D. Bierstedt, J. Bonet and A. Galbis [Mich. Math. J. 40, No. 2, 271-297 (1993; Zbl 0803.46023)] who considered the Banach space to be finite dimensional. A countable family $V$ of continuous, non-negative weights $v$ defined on $𝒰$ is used to construct a family of seminorms: if $f$ is a holomorphic function on $𝒰$ then

${p}_{v}\left(f\right):=sup\left\{v\left(x\right)|f\left(x\right)|:x\in 𝒰\right\}·$

The space $HV\left(𝒰\right)$ consists of those holomorphic functions for which ${p}_{v}\left(f\right)<\infty$ for all $v\in V$ and $H{V}_{0}\left(𝒰\right)$ is the subspace of those functions that vanish at infinity (this concept is defined suitably). The structure of these spaces under various conditions on the weights is investigated. Particular goals are conditions that (i) ensure that the $m$-homogeneous polynomials form a Schauder decomposition; and (ii) ensure reflexivity. A useful catalogue of 6 Examples is given. The final section discusses the existence of a predual for these spaces.

##### MSC:
 46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46B15 Summability and bases in normed spaces