As a consequence of Dvoretzky theorem " in any infinite dimensional Banach space $X$ and for any $\epsilon > 0$ and natural number $n$ there exists a subspace $L$ of $X$ with dim $L=n$ s.t. ${1\leq d(L,\mathbb{R}^n)< 1+\epsilon}$ where $d$ is the Banch-Mazur distance", Dvoretzky says that "An infinite dimensional Banach space $X$, which, for some $k$ > 1, has the property that all its $k$-dimensional subspaces are isometric to each other, is necessarily isometric to a Hilbert space."

Its proof to this consequence is

if we are given that for a fixed integer $k>1$ all the $k$-dimensional subspaces of a normed infinite dimensional linear space $L$ are isometric, it follows from Dvoretzky theorem that their common distance from $E^k$ cannot be positive; hence they are isometric to $E^k$ (by letting $\epsilon \rightarrow 0$) and the "parallelogram equality" $\|x+y\|^2+\|x-y\|^2=2(\|x\|^2+\|y\|^2)$ holds in $L$. Hence L is a pre-Hilbert space (inner product spaces). If $L$ is known to be complete we have thus proved.

I don't know how it follows from Dvoretzky theorem that if all $k$-dimensional subspaces, $k>1$, are isometric then they are isometric to $E^k$.

I have read the article "NORMED LINEAR SPACES EQUIVALENT TO INNER PRODUCT SPACES " by J. T. JOICHI, but i cann't connect the items. Any help, please. Thanks in advance.


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