Unconditional convergence

The unconditional convergence is a term used in functional analysis, which describes a particular convergence behavior of rows. One speaks of unconditional convergence of a series, when the convergence is insensitive to rearrangements of the series. In the finite this is equivalent to absolute convergence, in the infinite- is no longer the case.

Definition

Be a topological vector space. Be an index set and for all. It is said that a series converges absolutely against, if

  • The index set is countable and
  • Applies to any enumeration of the equation.

With

This term is most often studied in Banach spaces, but can be generally regarded as above in topological vector spaces in normed, or locally convex.

Applications

  • Using this definition can be, for example, introduce in a topological vector space, the usual notion of a " convergent sum of subspaces " as an extension of the already known sum of subspaces:
  • Sum of subspaces:
  • Extension " Convergent sum of subspaces ":
  • The Birkhoff integral for Banach space - valued functions is defined by means of unconditional convergence in Banach spaces.

Related to the absolute convergence

Riemann

Be the underlying Banach space and a countable index set. Then stating a theorem of Riemann that the series converges absolutely then exactly when it converges absolutely.

Set of Dvoretzky - Rogers

In infinite-dimensional spaces, the unconditional convergence and absolute convergence are no longer equivalent. This implies the theorem of Dvoretzky - Rogers, who was named after Aryeh Dvoretzky and Claude Ambrose Rogers. Precise, he states that in any infinite-dimensional Banach space an absolutely convergent series exist that are not absolutely convergent. The reversal, after every absolutely convergent series converges absolutely, also applies in the infinite-dimensional case.

Swell

  • Dirk Werner: Functional Analysis. 6, corrected edition, Springer -Verlag, Berlin 2007, ISBN 978-3-540-72533-6, page 232ff.
  • B. I. Golubov: Unconditional convergence. In: Michiel Hazewinkel (ed.): Encyclopaedia of Mathematics. Springer -Verlag, Berlin 2002, ISBN 1-4020-0609-8 ( online ).
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