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A separable Banach space $X$ is said to be finitely determined if for eachseparable 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 finitelyrepresentable in $X$. We also point out how a similar argument proves theKrivine-Maurey theorem on stable Banach spaces, and give the model theoreticinterpretations of some results.

Remarks on Banach spaces determined by their finite dimensional subspaces