Orlicz-Pettis theorem

The set of Orlicz-Pettis (after Władysław Orlicz and Billy James Pettis ) is a set of the mathematical sub-region of the functional analysis . In a certain situation, it allows to infer the weak convergence on the norm convergence in Banach spaces .

In infinite-dimensional Banach spaces, the weak topology is really weaker than the standard topology . For example, if the -th basis vector in Hilbert space , i. H. the sequence that has a 1 in the -th place and a 0 in all other places, then the sequence converges to 0 with respect to the weak topology. According to the Fréchet-Riesz representation theorem, every continuous linear functional has the form for a , and therefore applies . However, the sequence cannot converge with respect to the norm, because a possible norm limit would also have to be 0, but it applies to all indices . ${\ displaystyle e_ {n}}$ ${\ displaystyle n}$ ${\ displaystyle \ ell ^ {2}}$ ${\ displaystyle n}$ ${\ displaystyle (e_ {n}) _ {n}}$ ${\ displaystyle f \ in (\ ell ^ {2}) ^ {'}}$ ${\ displaystyle f (\ cdot) = \ langle \ cdot, \ xi \ rangle}$ ${\ displaystyle \ xi = (\ xi _ {k}) _ {k} \ in \ ell ^ {2}}$ ${\ displaystyle f (e_ {n}) = \ langle e_ {n}, \ xi \ rangle = \ xi _ {n} \ rightarrow 0}$ ${\ displaystyle (e_ {n}) _ {n}}$ ${\ displaystyle \ | e_ {n} \ | = 1}$ ${\ displaystyle n}$

The situation is the same for rows in Banach rooms. If one sets in the above example and for , so is . Therefore the series converges in the weak topology (towards 0) but not in the standard topology. ${\ displaystyle x_ {1} = e_ {1}}$ ${\ displaystyle x_ {n}: = e_ {n} -e_ {n-1}}$ ${\ displaystyle n> 1}$ ${\ displaystyle \ sum _ {n = 1} ^ {N} x_ {n} \, = \, e_ {N}}$ ${\ displaystyle \ sum _ {n} ^ {\ infty} x_ {n}}$

A number is teilreihenkonvergent if every part series converges, that is, if for every sequence converges. For series convergent series, the described difference between weak convergence and norm convergence no longer exists; that is precisely the content of the sentence presented here: ${\ displaystyle \ Sigma _ {n} x_ {n}}$ ${\ displaystyle \ Sigma _ {m} x_ {n_ {m}}}$ ${\ displaystyle n_ {1} <n_ {2} <n_ {3} <\ cdots}$

Orlicz-Pettis theorem :

A row in a Banach space that is partially convergent with respect to the weak topology is also partially convergent with respect to the standard topology .

This theorem was first proved by Orlicz in 1929 and independently of it by Pettis in 1938. Modern proofs use the Bochner integral . Conversely, the vector-valued integration theory was precisely the motivation for Pettis to deal with this sentence. This theorem has undergone a number of generalizations; one then speaks of theorems of the Orlicz-Pettis type. So z. B. in locally convex spaces that the partial series convergent series coincide with respect to the weak topology and with respect to the Mackey topology .

Individual evidence

↑ W. Orlicz: Contributions to the theory of orthogonal developments , Studia Math. Volume 1, (1929), pages 241-255
^ BJ Pettis: On Integration in Vector Spaces. Trans. Amer. Math. Soc. Volume 44 (1938), pages 277-304
^ Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5
^ P. Dierolf: Theorems of the Orlicz-Pettis Type for Locally Convex Spaces , Manuscripta Mathematica, Volume 20 (1977), pp. 73-94

[1] W. Orlicz: Contributions to the theory of orthogonal developments , Studia Math. Volume 1, (1929), pages 241-255

[2] BJ Pettis: On Integration in Vector Spaces. Trans. Amer. Math. Soc. Volume 44 (1938), pages 277-304

[3] Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5

[4] P. Dierolf: Theorems of the Orlicz-Pettis Type for Locally Convex Spaces , Manuscripta Mathematica, Volume 20 (1977), pp. 73-94