## 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
Full Text:

### References:

  Casazza, P.M., Shura, T.J.: Tsirelson Space, to appear.  Fabian, M., Whitfield, J.H.M., Zizler, V.:Norms with locally Lipschitzian derivatives. Israel J. Math.44, 262-276 (1983). · Zbl 0521.46009  Figiel, T.:Uniformly convex norms in spaces with unconditional basis. Seminaire Maurey-Schwartz 1974-75.  Figiel, T.:On the moduli of convexity and smoothness. Studia Math.56, 121-155 (1976). · Zbl 0344.46052  Figiel, T.: Lindenstrauss, J., Milman, V.:The dimension of almost spherical sections of convex bodies. Acta Math.139, 53-94 (1977). · Zbl 0375.52002  Krivine, J.L.:Finite dimensional subspaces of Banach lattices. Ann. Math.104, 1-29 (1976). · Zbl 0329.46008  Kwapien, S.:Isomorphic characterizations of inner product spaces by orthogonal series with vector coefficients. Studia Math.44, 583-595 (1972). · Zbl 0256.46024  Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer-Verlag. · Zbl 0403.46022  Makarov, B.M.:A characterization of Hilbert spaces, Math. Notes of the Academy of Sc. of the U.S.S.R.26, 863-867 (1979). · Zbl 0447.46015  ?:A condition for the isomorphism of a Banach space having the Orlicz property to a Hilbert space. J. of Soviet Math.27, 2514-2517 (1984). · Zbl 0546.46010  Maurey, B., Pisier, G.:Series de variables aléatoires vectorielles independantes et propriétés geometriques des espaces de Banach. Studia Math.58, 45-90 (1976). · Zbl 0344.47014  Meshkov, V.Z.:On smooth functions in the James spaces. Vestn. Mosk. Gos. Univ. Ser. Fiz.-Mat.29 (4), 9-13 (1974) (In Russian).  Meshkov, V.Z.:Smoothness properties in Banach spaces. Studia Math.63, 111-123 (1978). · Zbl 0416.46008  Milman, V., Pisier, G.:Banach spaces with a weak cotype 2 property, Israel J. Math.54, 139-158 (1986). · Zbl 0611.46022  Pisier, G.:Weak Hilbert space, preprint (1987).  Szankowski, A.:Subspaces without approximation property. Israel J. Math.30, 123-129 (1978). · Zbl 0384.46008  Szarek, S., Tomczak-Jaegermann, N.:On nearly Euclidean decompositions for some classes of Banach spaces. Compos. Math.40, 367-385 (1980). · Zbl 0432.46018  Tomczak-Jaegermann, N.:Banach-Mazur distances and finite dimensional operator ideals, Longman, to appear. · Zbl 0721.46004  Tsirelson, B.S.:Not every Banach space contains an imbedding of ? p or c 0. Funct. Anal. Appl.8 138-141 (1974). · Zbl 0296.46018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.