Remarks on Banach spaces determined by their finite dimensional subspaces

TL;DR

This paper investigates Banach spaces that are uniquely determined by their finite-dimensional subspaces, providing a direct proof of their properties and connecting to the Krivine-Maurey theorem.

Contribution

It offers a direct proof (avoiding model theory) that finitely determined Banach spaces contain all finitely representable subspaces, and relates this to the Krivine-Maurey theorem.

Findings

Finitely determined Banach spaces contain all finitely representable subspaces.

A direct proof approach without model theory is provided.

Connections to the Krivine-Maurey theorem on stable Banach spaces are established.

Abstract

A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without model theory) of the fact that every finitely determined space $X$ (isometrically) contains every (separable) space $Y$ which is finitely representable in $X$. We also point out how a similar argument proves the Krivine-Maurey theorem on stable Banach spaces, and give the model theoretic interpretations of some results.