Embedding normed linear spaces into C(X)

M. Fakhar; M. R. Koushesh; M. Raoofi

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arXiv · 1502.06008

Embedding normed linear spaces into C(X)

M. Fakhar·M. R. Koushesh·M. Raoofi

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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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BibTeX RIS

M. Fakhar, M. R. Koushesh, M. Raoofi. 2015-02-20. Embedding normed linear spaces into C(X). https://arxiv.org/abs/1502.06008

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