Phelps Property U and \(C(K)\) spaces

A subspace \(X\) of a Banach space \(Y\) has \(\textit{Property U}\) whenever every continuous linear functional on \(X\) has a unique norm-preserving (i.e., Hahn\(-\)Banach) extension to \(Y\) (Phelps, 1960). Throughout this document we introduce and develop a systematic study of the existence of \...

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Bibliographic Details
Published in:arXiv.org 2022-11
Main Authors: Cobollo, Ch, Guirao, A J, Montesinos, V
Format: Article
Language:English
Subjects:
Banach spaces
Continuity (mathematics)
Online Access:Get full text
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Summary:A subspace \(X\) of a Banach space \(Y\) has \(\textit{Property U}\) whenever every continuous linear functional on \(X\) has a unique norm-preserving (i.e., Hahn\(-\)Banach) extension to \(Y\) (Phelps, 1960). Throughout this document we introduce and develop a systematic study of the existence of \(\textit{U-embeddings}\) between Banach spaces \(X\) and \(Y\), that is, isometric embeddings of \(X\) into \(Y\) whose ranges have property U. In particular, we are interested in the case that \(Y=C(K)\), where \(K\) is a compact Hausdorff topological space. We provide results for general Banach spaces and for some specific set-ups, such as \(X\) being a finite-dimensional space or a \(C(K)\)-space.
ISSN:2331-8422