Two applications of smoothness in $C(K)$ spaces
Volume 225 / 2014
Studia Mathematica 225 (2014), 1-7
MSC: Primary 46B04, 46E15; Secondary 54H05.
DOI: 10.4064/sm225-1-1
Abstract
A simple observation about embeddings of smooth Banach spaces into $C(K)$ spaces allows us to construct a parametrization of the separable Banach spaces using closed subsets of the interval $[0,1]$. The same idea is applied to the study of the isometric embedding of $\ell _p$ spaces into certain $C(K)$ spaces with the additional condition that the functions of the image must be Lipschitz with respect to a fixed finer metric on $K$. The feasibility of that kind of embeddings is related to Szlenk indices.
