Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L2(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.

[1]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[2]  Gilles Pisier,et al.  Holomorphic semi-groups and the geometry of Banach spaces , 1982 .

[3]  V. Milman,et al.  Random subspaces of proportional dimension of finite dimensional normed spaces: Approach through the isoperimetric inequality , 1985 .

[4]  R. C. James.  Nonreflexive spaces of type 2 , 1978 .

[5]  Nicole Tomczak-Jaegermann,et al.  Projections onto Hilbertian subspaces of Banach spaces , 1979 .

[6]  A. Pajor,et al.  Subspaces of small codimension of finite-dimensional Banach spaces , 1986 .

[7]  B. Carl.  Entropy numbers, s-numbers, and eigenvalue problems , 1981 .

[8]  V. Milman,et al.  Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space , 1985 .

[9]  A. Pietsch,et al.  Weyl numbers and eigenvalues of operators in Banach spaces , 1980 .

[10]  Nicole Tomczak-Jaegermann,et al.  On nearly euclidean decomposition for some classes of Banach spaces , 1980 .

[11]  V. Milman.  Volume approach and iteration procedures in local theory of normed spaces , 1985 .

[12]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .

[13]  G. Pisier.  Quotients of Banach spaces of cotype $q$ , 1982 .

[14]  D. R. Lewis.  Ellipsoids defined by Banach ideal norms , 1979 .

[15]  G. Pisier,et al.  Un théorème sur les opérateurs linéaires entre espaces de Banach qui se factorisent par un espace de Hilbert , 1980 .