Embedding normed linear spaces into $C(X)$

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎University of Isfahan‎, ‎Isfahan 81745--163‎, ‎Iran‎, ‎and‎, ‎School of Mathematics‎, ‎Institute for Research in Fundamental Sciences (IPM)‎, ‎P.O‎. ‎Box: ‎19395--5746‎, ‎Tehran‎, ‎Iran.

2 Department of Mathematical Sciences‎, ‎Isfahan University of Technology‎, ‎Isfahan 84156--83111‎, ‎Iran‎, ‎and‎, ‎School of Mathematics‎, ‎Institute for Research in Fundamental Sciences (IPM)‎, ‎P.O‎. ‎Box‎: ‎19395--5746‎, ‎Tehran‎, ‎Iran.

3 Department of Mathematical Sciences‎, ‎Isfahan University of Technology‎, ‎Isfahan 84156--83111‎, ‎Iran.

Abstract

‎It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$‎. ‎Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology‎, ‎which is compact by the Banach--Alaoglu theorem‎. ‎We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone--Cech compactification of $L^*\setminus\{0\}$‎, ‎where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.

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Main Subjects

References

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History

• Receive Date: 12 July 2015
• Revise Date: 21 October 2015
• Accept Date: 21 February 2017