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


46B20 Geometry and structure of normed linear spaces
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