Eberlein – Šmulian's theorem

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The Eberlein-Šmulian Theorem (after William Frederick Eberlein and Witold Lwowitsch Smulian ) is a mathematical theorem from functional analysis , of a statement of compactness properties with respect to the weak topology of a Banach space makes.

In the topology , sequence compactness is examined as a variant of compactness. Neither of these two terms follows from the other. Let, for example, be the unit sphere in the dual space of the sequence space . With the weak - * - topology is compact according to the Banach-Alaoglu theorem . is not compact, because you look at the projections

so is a sequence in that has no convergent subsequence. Conversely, there are sequentially compact spaces that are not compact.

It is a well-known theorem that in metric spaces the terms compactness and sequence compactness coincide. Since the weak topology cannot be metrized on a Banach space, except in the finite-dimensional case, the following theorem by Eberlein – Šmulian is remarkable.

Eberlein – Šmulian's theorem

Let be a non-empty subset of a Banach space. Then the following statements are equivalent:

  • The weak end of is weak-compact (i.e. compact with respect to the weak topology).
  • The weak conclusion of is weakly sequence-compact (i.e. sequence-compact with respect to the weak topology).

Remarks

  1. For weakly closed subsets of a Banach space, the terms compact and consequential compact coincide.
  2. A comparable sentence for the weak - * - topology does not apply, as the example given above for the unit sphere in shows.
  3. In 1940 Šmulian showed that weakly compact sets in Banach spaces are weakly sequence compact. The reversal was only shown by Eberlein in 1947. This reversal was generalized in 1952 by Alexander Grothendieck to locally convex spaces , which are almost complete with regard to their Mackey topology .
  4. If a weakly compact subset is in a Banach space, it has the following peculiarity: A subset is weakly closed if and only if it is weakly sequence closed , i.e. H. if it also contains the limit value with every convergent sequence . It is clear that closed sets are sequentially closed. Conversely , if sequence-closed, then as a subset of the space which is sequence-compact according to the above sentence, sequence- compact and therefore, again according to Eberlein-Šmulian's theorem, weakly compact and thus weakly closed.
  5. Since a Banach space is reflexive if and only if its unit sphere is weakly compact, the Eberlein – Šmulian theorem gives another reflexivity criterion: A Banach space is reflexive if and only if its unit sphere is weakly sequential compact, and this is equivalent to that every bounded sequence has a weakly convergent subsequence.

literature

  • Joseph Diestel: Sequences and Series in Banach Spaces. Springer, New York et al. 1984, ISBN 0-387-90859-5 .