A normability condition for Frechet spaces
type: Item
instance of: scholarly article
title: A normability condition for Frechet spaces
zbMATH Open document ID: 0613.46003
DOI: 10.1007/BF01137474
author: Vladimir Kadets
author: M. I. Kadets
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let X be a Frechet space with a continuous norm p, and let \(U_ p=\{x:p(x)\leq 1\}.\) The authors prove that if every continuous linear functional on X which is bounded on \(U_ p\) attains its supremum on \(U_ p\), then p generates the original topology of X, which is therefore a reflexive Banach space. Now let V be a closed bounded absolutely convex subset of X, with associated seminorm p. A point \(x\in V=\{x:p(x)=1\}\) is said to be attainable if there exists a continuous linear functional f such that \(f(x)=1\) and f(y)\(\leq p(y)\) for all \(y\in V\). If V has non-empty interior, then by the Hahn-Banach Theorem every point of V is attainable. The authors prove a converse result: if V is in addition total, and every point of V is attainable, then V has non-empty interior, and hence X is a Banach space. Examples are given to show that metrizability cannot be dropped.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A35
zbMATH DE Number: 3992147
zbMATH Keywords: attainable point
zbMATH Keywords: Frechet space
zbMATH Keywords: Hahn-Banach Theorem
MaRDI profile type: Publication
cites work: Q3901969
cites work: Q3940289
cites work: Geometry of Banach spaces. Selected topics
cites work: Q4180773
cites work: A counterexample for a sup theorem in normed spaces
cites work: Q5736344
cites work: Q5573683
full work available at URL: https://doi.org/10.1007/bf01137474
OpenAlex ID: W2032006821
Recommended article: Q5748412
Recommended article: Q4289981
Recommended article: A Characterization of the Quasi-Normable Fréchet Spaces
Recommended article: Q3988897
Recommended article: On FrÉchet Spaces with a Dominant Norm
Recommended article: Fréchet differentiable norms on spaces of countable dimension
Recommended article: Some results on Fréchet spaces with the density condition
Recommended article: A characterization of normed spaces
Recommended article: Q3203543
Recommended article: Fréchet-differentiability of the norm in operator spaces
description: scientific article; zbMATH DE number 3992147