## A note on cotype of smooth spaces.(English)Zbl 0673.46004

A Banach space X is said to be of weak cotype 2 if there exist constants $$\delta\in (0,1)$$ and $$C>0$$ such that every finite-dimensional subspace E of X contains a subspace $$F\subset E$$ with dim $$F\geq \delta \dim E$$ and such that the Banach-Mazur distance $$d(F,\ell_ 2^{\dim F})\leq C$$. A Banach space X is said to be asymptotically Hilbertian if a constant $$C>0$$ exists such that for every positive integer n, there is a finite- dimensional subspace $$E_ n$$ of X such that for every n-dimensional subspace $$F_ n$$ of $$E_ n$$ we have that $$d(F_ n,\ell^ n_ 2)\leq C$$. A Banach space X is said to be a homogeneous space if all of its infinite dimensional subspaces are isomorphic to X. And, a Banach space X is $$C^ 2$$-smooth resp. $$LUC^ 2$$-smooth resp. $$LH^{2+a}$$-smooth (a$$\in (0,1])$$ if it admits a real valued function f with bounded nonempty support and such that the second order differential $$f''$$ of f is continuous, resp. locally uniformly continuous, resp. locally a- Hölder on X.
In this paper the two following results are proved:
(I) Let X be a separable Banach space. Then each one of the two following conditions implies that X is isomorphic to a Hilbert space:
(i) $$2+\sup (a>0$$; X is $$LH^{2+a}$$-smooth)$$>\inf (q$$; X is of cotype q).
(ii) X is $$LUC^ 2$$-smooth and asymptotically Hilbertian.
(II) Let X be a separable Banach space which is $$LH^{2+a}$$-smooth for some $$a>0$$. Then each one of the two following conditions implies that X is isomorphic to a Hilbert space.
(i) X is of weak cotype 2.
(ii) X is a homogeneous space.
Reviewer: Yu Xin tai

### MSC:

 46B20 Geometry and structure of normed linear spaces
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### References:

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