Weak convergence

The title of this article is ambiguous. See Weak Convergence (disambiguation) for other meanings .

The weak convergence is a concept of convergence in the functional analysis , a branch of mathematics . The weak convergence is defined on standardized spaces and provides there, for example, more general criteria for the existence of minima and maxima than the convergence with regard to the norm of the underlying space.

The weak convergence is closely related to the weak topology and in some cases corresponds to the convergence on this topology. However, it can happen that the characterization of topological properties by sequences (which happens in the case of weak convergence) does not coincide with the purely topological characterization (as it happens in the case of the weak topology). So it is possible that closed sets in the weak topology are not closed with weak consequences .

definition

Given a normalized space and its topological dual space , i.e. the vector space of all continuous linear functionals ${\displaystyle X}$ ${\displaystyle X'}$

${\displaystyle x'\colon X\to \mathbb {K} }$.

A sequence in is then called weakly convergent to (in ) if ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$

${\displaystyle \lim _{n\to \infty }x'(x_{n})=x'(x)}$ for all ${\displaystyle x'\in X'}$

applies.

If the sequence converges weakly against , one writes or also or respectively . For a clearer definition of weak convergence, the convergence is with respect to the standard of known then also strong convergence or norm convergence. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x}$ ${\displaystyle x_{n}\rightharpoonup x}$ ${\displaystyle x_{n}{\stackrel {\sigma }{\rightarrow }}x}$ ${\displaystyle \textstyle \sigma {\text{-}}\!\lim _{n\to \infty }x_{n}=x}$ ${\displaystyle X}$

example

Considering as a normed space the L ^p -space with , due to the so is duality of L ^p -spaces the dual space norm isomorphic to , wherein the to conjugated index is. So it applies . ${\displaystyle X}$ ${\displaystyle L^{p}}$ ${\displaystyle p\in (1,\infty )}$ ${\displaystyle X'}$ ${\displaystyle L^{q}}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

Hence every continuous linear functional

${\displaystyle x'\colon X\to \mathbb {R} }$

a representation of the shape

${\displaystyle x'(f)=\int fg\,\mathrm {d} \mu }$,

where and is. Thus a sequence of functions from is weakly convergent to if and only if ${\displaystyle g\in L^{q}}$ ${\displaystyle f\in L^{p}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle L^{p}}$ ${\displaystyle f\in L^{p}}$

${\displaystyle \lim _{n\to \infty }\int f_{n}g\,\mathrm {d} \mu =\int fg\,\mathrm {d} \mu {\text{ für alle }}g\in L^{q}}$

applies. This is precisely the weak convergence in L ^p .

Basic properties

Uniqueness

The limit of weakly convergent sequences is clearly determined. This follows from the fact that the dual space separates , which means: ${\displaystyle X}$

Are out , there is a with . ${\displaystyle x\neq y}$ ${\displaystyle X}$ ${\displaystyle x'\in X'}$ ${\displaystyle x'(x)\neq x'(y)}$

This is a consequence of Hahn-Banach's theorem .

Narrow-mindedness

Weakly convergent sequences are always constrained in . For if it converges weakly, the consequences are limited in for all . According to a corollary of the Banach-Steinhaus theorem, this is equivalent to the boundedness of . ${\displaystyle X}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x'\in X'}$ ${\displaystyle (x'(x_{n}))_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$

Naming topological properties

Topological properties that are defined by weak convergence are usually identified by the prefix "weakly follow-". That's a lot ${\displaystyle M}$

• weakly sequentially completed when the limit value of each weak convergent sequence in again is ${\displaystyle M}$ ${\displaystyle M}$
• weak sequence compact if each sequence has a weakly convergent partial sequence whose limit is again in . ${\displaystyle M}$ ${\displaystyle M}$

This designation applies to all topological properties that can be defined using sequences. Another example of this would be the weak relative sequence compactness .

In general, these terms do not coincide with the corresponding purely topological terms in the weak topology (seclusion, compactness, relative compactness, etc.). For details, see #relation to weak topology

Relationship to norm convergence

The weak convergence always follows from the norm convergence. Because is convergent with respect to the norm, then applies ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x}$

${\displaystyle \lim _{n\to \infty }x'(x_{n})=x'(x)}$

for everyone , because this is exactly the continuity required by them. In general, the reverse does not apply, so there may be weakly convergent sequences that are not norm-convergent. The set of Mazur provides a limited reversal. It says that from the terms of a weakly convergent sequence a second sequence can always be constructed by convex combinations , which converges with respect to the norm. ${\displaystyle x'\in X'}$ ${\displaystyle x'}$

An example of a weakly convergent sequence that is not norm-convergent can be constructed in the sequence space , where is. One chooses as a consequence ${\displaystyle \ell ^{p}}$ ${\displaystyle p\in (1,\infty )}$

${\displaystyle e_{1}=(1,0,0,\dots ),\,e_{2}=(0,1,0,0,\dots ),\dots }$,

so is always

${\displaystyle \lim _{n\to \infty }\|e_{n}\|_{\ell ^{p}}=1}$.

However , there are a series of so ${\displaystyle \Phi \in \left(\ell ^{p}\right)'}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \ell ^{q}}$

${\displaystyle \Phi (x)=\sum _{i=1}^{\infty }a_{i}x_{i}}$

is. Here again is the index to be conjugated. So is ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle \lim _{n\to \infty }\Phi (e_{n})=\lim _{n\to \infty }a_{n}=0}$,

since is a null sequence. Thus the sequence converges weakly to 0, but not to 0 with respect to the norm. ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

In particular, the standard is no longer steadily with respect to the weak convergence, but only lower continuous . So if a sequence converges weakly in against , then we have ${\displaystyle \|\cdot \|}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle x}$

${\displaystyle \|x\|\leq \liminf _{n\to \infty }\|x_{n}\|}$.

Relationship to weak topology

In metric spaces , many topological properties can be characterized in two equivalent ways: Either via sequences and their properties or via the properties of the induced topology. An example of this is the closure: either closed sets are characterized as those sets in which the limit value of a convergent sequence is contained in the set, or as the complement of an open set.

The two above characterizations are also still possible in general topological spaces , but the terms obtained then generally no longer agree with one another. The terms obtained from the sequences are then given the prefix "Sequence-" (sequence closed, sequence compact, etc.)

As already mentioned above, topological terms obtained via the weak convergence are given the prefix "weakly follow". The terms obtained from the weak topology then correspond to the classic topological characterization and get along with the prefix "weak". Since the weak topology is generally not metrizable , the two types of characterization differ. Therefore, they have to be distinguished in general. Statements that provide the equivalence of the two characterizations are often independent sentences. One of them is, for example, the Eberlein – Šmulian theorem , which establishes the equivalence of compactness and sequence compactness with regard to the weak topology on Banach spaces.

It should be noted that in addition to the weak convergence, there is also a convergence in the weak topology. Because of the reasons mentioned above, this is generally different from weak convergence and must be formalized via networks or filter convergence .

Relationship to weak - * convergence

The weak convergence can easily be transferred to the dual space . Denotes the bidual space , then weakly converges to in , if ${\displaystyle X'}$ ${\displaystyle X''}$ ${\displaystyle (x'_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x'}$ ${\displaystyle X'}$

${\displaystyle \lim _{n\to \infty }x''(x'_{n})=x''(x')}$for everyone . ${\displaystyle x''\in X''}$

In the dual space the weak - * - convergence can also be defined: A sequence is called weak - * - convergent to if ${\displaystyle (x'_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x'\in X'}$

${\displaystyle \lim _{n\to \infty }x_{n}'(x)=x'(x)}$for everyone . ${\displaystyle x\in X}$

Denoting the canonical map into the Bidualraum , a sequence converges if and weak against in when the sequence low- * compared to converge. Furthermore, the weak convergence in always follows from the weak convergence in . Both statements essentially follow from the properties of the canonical mapping. If the space is reflexive , then weak convergence in and weak - * - convergence in even coincide. ${\displaystyle J_{X}}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle (J_{X}(x_{n}))_{n\in \mathbb {N} }}$ ${\displaystyle J_{X}(x)}$ ${\displaystyle X''}$ ${\displaystyle X'}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X'}$

Reflexive spaces and weak convergence

In reflexive spaces , stronger statements apply to the weak convergence. This is based on the fact that, by definition, the mapping which, among other things, links the weak convergence in with the weak - * - convergence in is additionally surjective . Thus in a reflexive space every restricted sequence has a weakly convergent subsequence. As mentioned above, weak convergence in and weak - * - convergence in also match in reflexive spaces . ${\displaystyle J_{X}}$ ${\displaystyle X}$ ${\displaystyle X''}$ ${\displaystyle X'}$ ${\displaystyle X'}$

Weak convergence in Hilbert spaces

In a Hilbert space , weak convergence is equivalent to boundedness and component-wise convergence with respect to an orthogonal basis . Since every Hilbert space is reflexive, a bounded sequence in a Hilbert space always has a weakly convergent subsequence.

literature

• Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .
• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .

Individual evidence

1. ^ Werner: Functional Analysis. 2011, p. 405.
2. ^ Werner: Functional Analysis. 2011, p. 106.
3. ↑ Old: Linear functional analysis. 2012, p. 237.
4. ↑ von Querenburg: Set theoretical topology. 2011, p. 75.
5. ↑ Old: Linear functional analysis. 2012, p. 238.
6. ↑ Old: Linear functional analysis. 2012, p. 245.