Is there a separable, reflexive Banach space $Z$ such that for every finite dimensional space $X$ and every $a>0$, there is a $1+a$-embedding of $X$ into $Z$?
I can do the question without the 'reflexive' (in which case it's true), but I'm totally stuck on how to find a reflexive space with this property.
Please help? Thanks.
On
I thought I would mention a different answer to Norbert's since the paper containing the result I cite is not that well known and deserves to be advertised. Szankowski has shown that there exists a sequence of Banach spaces $X_m$, $m\in\mathbb{N}$, each isomorphic to $\ell_2$, with the following property: every finite dimensional Banach space is isometrically isomorphic to a contractively complemented subspace of $(\bigoplus_{m\in\mathbb{N}}X_m)_{\ell_2}$.
The paper of Szankowski is An example of a universal Banach space, Israel Journal of Mathematics 11 (1972), 292-296.
As an aside: When I was a postdoc in France a couple of years ago, some of the Banach space experts there thought it was an open problem whether a separable Banach space containing every finite dimensional Banach space isometrically necessarily contained a subspace isomorphic to $C([0,1])$. Szankowski's result (published almost 40 years earlier!) obviously shows that the answer to this question is negative.
This is so called Johnson's space $C_p$. For explicit construction see p. 71 in Tensor Norms and Operator Ideals. A. Defant, K. Floret.