Dunford–Pettis property

The Dunford - Pettis property (after N. Dunford and BJ Pettis ) is considered in the mathematical branch of functional analysis property of Banach spaces.

Definition

The following definition is due to A. Grothendieck (1953 ):

A Banach space has the Dunford - Pettis property if every weakly compact linear operator is already compact for every Banach space.

Examples

  • The sequence spaces, and have the Dunford - Pettis property, the sequence spaces are not.
  • Is a finite measure space, then L1 has the Dunford - Pettis property. That this is the case, was previously proved by N. Dunford and BJ Pettis and Grothendieck was for the motivation for naming.
  • Is a compact Hausdorff space, so has the Banach space of continuous functions the Dunford - Pettis property, as was proved by Grothendieck.
  • No infinite-dimensional reflexive Banach space has the Dunford - Pettis property.

A characterization

For a Banach space the following are equivalent:

  • Has the Dunford - Pettis property.
  • Is a suite in a weak limit and a sequence in the dual space with a weak limit, then for.
  • Is a suite in a weak limit and a sequence in the dual space with a weak limit, then for.

Properties

Has the dual space of the Banach space the Dunford - Pettis property, as well.

Since the commutative C * - algebra of the form is a compact Hausdorff space (see Gelfand - Neumark ) after the revision mentioned in the examples set by the Dunford - Pettis Grothendieck property. Since and ( see Article sequence space ), it follows that also Dunford - Pettis and the property.

Swell

  • Robert E. Megginson: An Introduction to Banach Space Theory. Springer, New York 1998, ISBN 0-387-98431-3.
  • Joseph Diestel: Sequences and Series in Banach Spaces. Springer, New York, Berlin, Heidelberg, Tokyo 1984, ISBN 0-387-90859-5.
  • Functional Analysis
Nelson Dunford Billy James Pettis Alexander Grothendieck Weak topology C*-Algebra
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