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The dual form of the approximation property for a Banach space and a subspace

100%

Figiel T., Johnson W.

Studia Mathematica

2015

tom 231

nr 3

287-292

Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. In particular, if the pair (X,Y) has the AP then X, Y, and the quotient space X/Y have the classical Grothendieck AP. The main result is an easy to apply dual formulation of this property. Applications are given to three-space properties; in particular, if X has the approximation property and its subspace Y is $𝓛_{∞}$, then X/Y has the approximation property.

Stochastic approximation properties in Banach spaces

81%

Fonf V., Johnson W., Pisier G., Preiss D.

Studia Mathematica

2003

tom 159

nr 1

103-119

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.

Ograniczanie wyników

2 Studia Mathematica

2 Johnson W.

1 Figiel T.

1 Fonf V.

1 Pisier G.

1 Preiss D.

1 2015

1 2003

Wyniki wyszukiwania

The dual form of the approximation property for a Banach space and a subspace

Stochastic approximation properties in Banach spaces