Separable, reflexive space that is almost isometrically universal for the class of finite dimensional normed spaces.

Question

My question is whether there is a separable, reflexive Banach space $X$ s.t. for any finite dimensional normed space $E$ and ${\epsilon>}0$, there is a linear isomorphism $T$ from $E$ to its image in $X$ s.t $||T|| ||T^{-1}||<1+\epsilon$?

For instance it is relatively straightforward to show that $c_0$ is almost ismoertically universal for finite dimensional normed spaces (consider ${\delta}$-nets for the closed unite sphere in a given $E$ and projections) but I have no idea how to approach this case.

Would $\Bigl( \sum_{n=1}^\infty \ell_\infty^n\Bigr)_2$ work? — David Mitra, Nov 02 '15 at 23:26

score 1 · Accepted Answer · answered Nov 03 '15 at 11:03

1

A good example of such space is a Johnson's $C_p$-space. For details see page 71 in Tensor Norms and Operator Ideals. A. Defant, K. Floret.

answered Nov 03 '15 at 11:03

Norbert

56,803

Separable, reflexive space that is almost isometrically universal for the class of finite dimensional normed spaces.

1 Answers1