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A FURTHER PROPERTY OF SPHERICAL ISOMETRIES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  15 May 2014

RYOTARO TANAKA
RYOTARO TANAKA*
Affiliation:
Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan email [email protected]
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Abstract

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In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$-dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$-extreme points of the unit ball of $X$ onto that of $Y$.

Keywords

Tingley’s problemisometric extension problemframek-extreme point

MSC classification

Secondary: 46B04: Isometric theory of Banach spaces 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 304 - 310
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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