Geodesic
Stochastic approximation properties in Banach spaces
V. P. Fonf 1   ; W. B. Johnson 2   ; G. Pisier 3   ; D. Preiss 4
1 Ben-Gurion University of the Negev P.O. Box 653 Beer-Sheva 84105, Israel
2 Department of Mathematics Texas A&M University Department of Mathematics College Station, TX 77843, U.S.A.
3 Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A. and Equipe d'Analyse, Case 186 Université Paris VI 75252 Paris, Cedex 05, France
4 Department of Mathematics University College London London, Great Britain
Studia Mathematica, Tome 159 (2003) no. 1, pp. 103-119 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We show that a Banach space $X$ has the stochastic approximation property iff it has the stochasic basis property, and these properties are equivalent to the approximation property if $X$ has nontrivial type. If for every Radon probability on $X$, there is an operator from an $L_p$ space into $X$ whose range has probability one, then $X$ is a quotient of an $L_p$ space. This extends a theorem of Sato's which dealt with the case $p=2$. In any infinite-dimensional Banach space $X$ there is a compact set $K$ so that for any Radon probability on $X$ there is an operator range of probability one that does not contain $K$.
DOI : 10.4064/sm159-1-5
Keywords: banach space has stochastic approximation property has stochasic basis property these properties equivalent approximation property has nontrivial type every radon probability there operator space whose range has probability quotient space extends theorem satos which dealt infinite dimensional banach space there compact set radon probability there operator range probability does contain

V. P. Fonf  1   ; W. B. Johnson  2   ; G. Pisier  3   ; D. Preiss  4

1 Ben-Gurion University of the Negev P.O. Box 653 Beer-Sheva 84105, Israel
2 Department of Mathematics Texas A&M University Department of Mathematics College Station, TX 77843, U.S.A.
3 Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A. and Equipe d'Analyse, Case 186 Université Paris VI 75252 Paris, Cedex 05, France
4 Department of Mathematics University College London London, Great Britain 
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V. P. Fonf; W. B. Johnson; G. Pisier; D. Preiss. Stochastic approximation properties
 in Banach spaces. Studia Mathematica, Tome 159 (2003) no. 1, pp. 103-119. doi: 10.4064/sm159-1-5

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