A Bounded Closed Convex Set in $c_0(\mathbb{C})$ without Modulus Support Functionals

We construct a bounded closed convex set in $c_0(\mathbb{C})$ on which no nonzero continuous complex-linear functional attains its maximum modulus. Consequently, this answers in the negative the two longstanding questions raised by Deville, Godefroy and Zizler in \cite[p. 35]{DGZ}. The construction uses a small perturbation of the identity on $c_0(\mathbb{R})$.

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-27
DOI
https://doi.org/10.48550/arxiv.2610.36771
Primary Topic
Advanced Banach Space Theory
Type
preprint
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preprint

A Bounded Closed Convex Set in $c_0(\mathbb{C})$ without Modulus Support Functionals

Xin Luo
arXiv (Cornell University)
Advanced Banach Space Theory
preprint

A Bounded Closed Convex Set in $c_0(\mathbb{C})$ without Modulus Support Functionals

Xin Luo
preprint en

Abstract

We construct a bounded closed convex set in $c_0(\mathbb{C})$ on which no nonzero continuous complex-linear functional attains its maximum modulus. Consequently, this answers in the negative the two longstanding questions raised by Deville, Godefroy and Zizler in \cite[p. 35]{DGZ}. The construction uses a small perturbation of the identity on $c_0(\mathbb{R})$.

arXiv (Cornell University)
Advanced Banach Space Theory
DOIOpen Access PDFOpenAlex
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A Bounded Closed Convex Set in $c_0(\mathbb{C})$ without Modulus Support Functionals — Xin Luo · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS