LFC bumps on separable Banach spaces
type: Item
instance of: scholarly article
title: LFC bumps on separable Banach spaces
zbMATH Open document ID: 1193.46003
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: Let \(X\) be a Banach space. A function \(b:X\to \mathbb{R}\) is a bump function if it has bounded and nonempty support. The function \(b\) depends locally on finitely many coordinates (LFC for short) if for each \(x\in X\) there are a neighborhood \(U\) of \(x\) and a finite subset \(F\subset X^*\) such that \(b(y)=b(z)\) whenever \(y,z\in U\) such that \(f(y)=f(z)\) for all \(f\in F\). A norm is said to be LFC if it is LFC away from the origin. The LFC notion is closely related to the class of polyhedral Banach spaces. \textit{P. Hájek} [Proc. Am. Math. Soc. 123, 3817--3821 (1995; Zbl 0853.46010)] characterized them as those admitting an equivalent \(\mathcal{C}^{\infty}\)-smooth and LFC norm. It remains an open problem whether every separable Banach space with a \(\mathcal{C}^{\infty}\)-smooth LFC bump is polyhedral. In the paper under review, the authors construct a \(\mathcal{C}^{\infty}\)-smooth LFC bump function in every separable Banach space admitting a continuous, LFC bump function. This result answers a problem posed by several authors (see, e.g., [\textit{P. Hájek, M. Johanis}, J. Math. Ann. Appl. 338, 1131--1139 (2008; Zbl 1136.46005)]).
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 5673281
zbMATH Keywords: Smooth bump functions
zbMATH Keywords: locally depending on finitely many coordinates
zbMATH Keywords: polyhedral Banach spaces
author: M. Jiménez-Sevilla
author: Luis Sánchez González
reviewed by: A. M. Plichko
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.10.049
OpenAlex ID: W2062416591
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DOI: 10.1016/J.JMAA.2009.10.049
description: scientific article; zbMATH DE number 5673281