, Volume 22, Issue 6, pp 1721-1724
Date: 10 Apr 2006

The Isometric Extension of an Into Mapping from the Unit Sphere \( S{\left( {{\ell }^{\infty }_{{{\left( 2 \right)}}} } \right)} \) to \( S{\left( {L^{1} {\left( \mu \right)}} \right)} \)

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Abstract

This is the first paper to consider the isometric extension problem of an into–mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into–isometric mapping from the unit sphere \( S{\left( {{\ell }^{\infty }_{{{\left( 2 \right)}}} } \right)} \) to \( S{\left( {L^{1} {\left( \mu \right)}} \right)} \) can be (real) linearly isometrically extended.

This paper is supported by The National Natural Science Foundation of China (10571090) and The Research Fund for the Doctoral Program of Higher Education (20010055013)