Isometric embeddings of compact spaces into Banach spaces

TL;DR

This paper constructs a compact metric space that, when embedded isometrically into a Banach space, ensures the space contains isometric copies of all separable Banach spaces, and explores conditions for embedding specific subsets.

Contribution

It demonstrates the existence of a universal compact space for isometric embeddings of all separable Banach spaces and addresses embedding conditions for specific subsets.

Findings

Existence of a compact space embedding all separable Banach spaces

Positive results for embedding unit balls of certain spaces

Conditions under which subsets imply the presence of the entire space

Abstract

We show the existence of a compact metric space $K$ such that whenever $K$ embeds isometrically into a Banach space $Y$, then any separable Banach space is linearly isometric to a subspace of $Y$. We also address the following related question: if a Banach space $Y$ contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space $X$, does it necessarily contain a subspace isometric to $X$? We answer positively this question when $X$ is a polyhedral finite-dimensional space, $c_0$ or $\ell_1$.