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NONEXPANSIVE MAPPINGS AND EXPANSIVE MAPPINGS ON THE UNIT SPHERES OF SOME F-SPACES

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  22 April 2010

DONG-NI TAN
DONG-NI TAN*
Affiliation:
School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, PR China (email: 0110127@mail.nankai.edu.cn)
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Abstract

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This paper gives a characterization of nonexpansive mappings from the unit sphere of ℓβ (Γ) onto the unit sphere of ℓβ (Δ) where 0<β≤1. By this result, we prove that such mappings are in fact isometries and give an affirmative answer to Tingley’s problem in ℓβ (Γ) spaces. We also show that the same result holds for expansive mappings between unit spheres of ℓβ (Γ) spaces without the surjectivity assumption.

Keywords

nonexpansive mappingexpansive mappingisometric extensionTingley’s problemℓβ(Γ) spaces

MSC classification

Secondary: 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 22 - 30
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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