1. On C_{0}(S, X)-distortion of the class of all separable Banach spaces.
- Author
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Galego, Elói Medina and Silva, André Luis Porto da
- Subjects
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BANACH spaces , *HAUSDORFF spaces , *EMBEDDING theorems , *COMMERCIAL space ventures , *COMPACT spaces (Topology) - Abstract
Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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