×

Cotype for p-Banach spaces. (English) Zbl 0606.46011

Let \(X\) be a real vector space, \(0<p\leq 1\) and a p-convex norm on \(X\), i.e. a function \(\| \cdot \|: X\to R_+\) with the properties:
i) \(\| x\| >0\), for \(x\neq 0\)
ii) \(\| ax\| =| a| \| x\|\), \(a\in {\mathbb{R}}\) and \(x\in X\)
iii) \(\| x+y\|^ p\leq \| x\|^ p+\| y\|^ p\), \(x,y\in X.\)
Then \((X,\| \cdot \|)\) is a p-Banach space. And a p-Banach space \((X,\| \cdot \|)\) is of cotype \(q\), \(0<q\leq +\infty\), if there exists a constant \(C>0:\) \((\sum^{n}_{1}\| x_ i\|^ q)^{1/q}\leq C\int^{1}_{0}\| \sum^{n}_{1}r_ i(t)x_ i\| dt\) for all \(n\in {\mathbb{N}}\) and \(x_ i\in X\), where \(\{r_ i(t)\}^ n_ 1\) are the Rademacher functions on \([0,1]\).
Now in this paper the theorem of Maurey and Pisier for the cotype in p- Banach spaces is proved. Precisely it is proved that for \((X,\| \cdot \|)\) a real p-Banach space and \(q(X)=\inf \{q: X\) is of \(q\)-Rademacher cotype\(\}\ell^{q(X)}\) is finitely representable in \(X\), when \(q(X)\) is real and, for each \(\epsilon >0\), \(c_ 0\) is \((2^{1/p-1}+\epsilon)\) finitely representable in X, when \(q(X)=\infty\).
Reviewer: K.Stathakopoulos

MSC:

46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] J. Bastero : A restricted form of the theorem of Maurey-Pisier for the cotype in p-Banach spaces . Anns. Ins. Henri Poincaré -B-XVIII (3) (1982), 293-304. · Zbl 0486.46016
[2] A. Brunel and L. Sucheston : On J-convexity and ergodic super-properties of Banach spaces . Trans. A.M.S. 204 (1975) 79-90. · Zbl 0273.46013 · doi:10.2307/1997350
[3] J.P. Kahane : Series of random functions . Heath. Math. Monog. Mass.: Health and Co. 1968. · Zbl 0192.53801
[4] N.J. Kalton : The convexity type of quasi-Banach spaces . (unpublished paper). · Zbl 0408.46007
[5] J.L. Krivine : Sous-espaces the dimensión finie des espaces de Banach réticulés . Annals of Math. 104 (1976) 1-29. · Zbl 0329.46008 · doi:10.2307/1971054
[6] B. Maurey and G. Pisier : Séries de variables aléatoires vectorielles indépendantes et propietés géométriques des espaces de Banach. Studia Math. 58 (1976) 45-90. · Zbl 0344.47014
[7] V. Milman and M. Sharir : A new proof of the Maurey-Pisier Theorem . Israel J. of Math. 33 (1979) 73-87. · Zbl 0418.46010 · doi:10.1007/BF02760534
[8] G. Pisier : On the dimension of the lnp-subspaces of Banach spaces, for 1 \leq p < 2 . TAMS 276 (1983) 201-212. · Zbl 0509.46016 · doi:10.2307/1999427
[9] Z. Uriz : Una nota sobre espacios p-Banach isomorfos a c0 . Actas IX Jornadas Matemáticas Hispano-Lusas, 1. Univ. Salamanca (1982) 405-408.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.