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## Stochastic approximation properties in Banach spaces

### Tom 159 / 2003

Studia Mathematica 159 (2003), 103-119 MSC: Primary 46A35, 46B25, 46E30, 60B11. DOI: 10.4064/sm159-1-5

#### Streszczenie

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 $K$.

#### Autorzy

• V. P. FonfBen-Gurion University of the Negev
P.O. Box 653
Beer-Sheva 84105, Israel
e-mail
• W. B. JohnsonDepartment of Mathematics
Texas A&M University
Department of Mathematics
College Station, TX 77843, U.S.A.
e-mail
• G. PisierDepartment 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
e-mail
e-mail
• D. PreissDepartment of Mathematics
University College London
London, Great Britain
e-mail

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