A two-dimensional Hahn-Banach theorem
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- by B. L. Chalmers and B. Shekhtman PDF
- Proc. Amer. Math. Soc. 129 (2001), 719-724 Request permission
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 $\|T\|=\|\tilde T\|$ 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 $\|T\|=\|\tilde T\|$, there exists a choice of basis $v=v_1, v_2$ such that $\|u_i\| = \|\tilde { u}_i\|,\;i=1,2$; i.e., each $u_i$ is a Hahn-Banach extension of $\tilde { u}_i$.References
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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
- MR Author ID: 195882
- Email: boris@math.usf.edu
- Received by editor(s): January 5, 1999
- Published electronically: November 8, 2000
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1801997