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A two-dimensional Hahn-Banach theorem


Authors: B. L. Chalmers and B. Shekhtman
Journal: Proc. Amer. Math. Soc. 129 (2001), 719-724
MSC (2000): Primary 46B20; Secondary 47A20
DOI: https://doi.org/10.1090/S0002-9939-00-05944-X
Published electronically: November 8, 2000
MathSciNet review: 1801997
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Abstract:

Let $\tilde T=\sum_{i=1}^n \tilde{ u}_i\otimes v_i:\;V\rightarrow V=[v_1,...,v_n]\subset X$, where $\tilde{ u}_i\in V^{*}$ and $X$ is a Banach space. Let $T= \sum_{i=1}^n u_i\otimes v_i:\;X\rightarrow V$ be an extension of $\tilde T$ to all of $X$ (i.e., $u_i\in X^{*}$) such that $T$ has minimal (operator) norm. In this paper we show in particular that, in the case $n=2$and the field is R, there exists a rank-$n$ $\tilde T$ such that $\Vert T\Vert=\Vert\tilde T\Vert$ for all $X$ if and only if the unit ball of $V$ is either not smooth or not strictly convex. In this case we show, furthermore, that, for some $\Vert T\Vert=\Vert\tilde T\Vert$, there exists a choice of basis $v=v_1,\, v_2$such that $\Vert u_i\Vert = \Vert\tilde{ u}_i\Vert,\;i=1,2$; i.e., each $u_i$ is a Hahn-Banach extension of $\tilde{ u}_i$.


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Additional Information

B. L. Chalmers
Affiliation: Department of Mathematics, University of California, Riverside, California 92507
Email: blc@math.ucr.edu

B. Shekhtman
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
Email: boris@math.usf.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05944-X
Received by editor(s): January 5, 1999
Published electronically: November 8, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society