Weak topology

The weak topology is a special topology and is situated in the border area of ​​the two mathematical sub-areas of topology and functional analysis . It is defined on standardized spaces or, more generally, on locally convex Hausdorff spaces .

The weak topology is closely related to the weak convergence . 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

A locally convex Hausdorff space such as a standardized space , provided with the standard topology, is given . Let be the dual space of . ${\displaystyle (X,\tau _{0})}$ ${\displaystyle X'}$ ${\displaystyle X}$

The weak topology can be defined in two equivalent ways: either as an initial topology or by specifying a zero neighborhood base .

On access as initial topology, the weak topology on than the initial topology on respect defined. Therefore it is the coarsest topology on , so that all are continuous. It has the sub-base as the initial topology ${\displaystyle \tau _{w}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle x'\in X'}$

${\displaystyle {\mathcal {S}}_{w}:=\{{x'}^{-1}(O)\mid O\subset \mathbb {K} {\text{ offen }},\;x'\in X'\}}$

and is clearly determined by this.

For access using a zero environment base, one defines

${\displaystyle U_{F,\varepsilon }=\{x\in X\mid x'(x)\leq \varepsilon {\text{ für alle }}x'\in F\}}$,

where is here . The weak topology is then the topology on the basis of the null neighborhood ${\displaystyle F\subset X'}$ ${\displaystyle X}$

${\displaystyle {\mathcal {U}}_{0}=\{U_{F,\varepsilon }\mid F\subset X',\;|F|{\text{ endlich }},\varepsilon >0\}}$

and is clearly determined by this.

Open sets in the weak topology

Depending on the definition, the open sets are constructed differently in the weak topology.

When constructing the initial topology, the sub-base of the weak topology specified in the definition is first created . It consists of archetypes of open sets in among the elements of . All sets in are open in the weak topology. Then one forms the set of all cuts of finitely many sets from the sub-basis : ${\displaystyle {\mathcal {S}}_{w}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle X'}$ ${\displaystyle {\mathcal {S}}_{w}}$ ${\displaystyle {\mathcal {S}}_{w}}$

${\displaystyle {\mathcal {B}}_{w}:=\{A\subset X\mid A{\text{ ist Schnitt endlich vieler Mengen aus }}{\mathcal {S}}_{w}\}}$.

${\displaystyle {\mathcal {B}}_{w}}$then forms a basis of the weak topology and all sets are then open with regard to the weak topology. The weak topology itself then consists of all sets that are a union of (any number of) sets . ${\displaystyle {\mathcal {B}}_{w}}$ ${\displaystyle {\mathcal {B}}_{w}}$

In the construction on the basis of zero neighborhoods, one exploits that a set is open if and only if it is neighborhood of each of its points. So then applies

${\displaystyle A}$is open in the weak topology for all one exists , so that is. ${\displaystyle \iff }$ ${\displaystyle x\in A}$ ${\displaystyle U\in {\mathcal {U}}_{0}}$ ${\displaystyle x+U\subset A}$

This exploits, on the one hand, that a set is a neighborhood of a point if and only if it contains a set of the neighborhood base of , and that the neighborhood base of corresponds exactly to in the case of the weak topology . ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle x}$ ${\displaystyle x+{\mathcal {U}}_{0}}$

properties

• The weak topology makes it a locally convex space . ${\displaystyle X}$
• The closed unit sphere of is weakly compact if and only if is a reflexive Banach space . ${\displaystyle X}$ ${\displaystyle X}$
• In locally convex topological vector spaces, closed and convex subsets are weakly closed.
• The Eberlein-Šmulian Theorem provides the equivalence of compactness and sequential compactness respect. The weak topology on Banach spaces fixed.

Designations and notation

For a more precise delimitation from the weak topology , the topology is also referred to as the initial topology , in the case of a standardized space also as the original topology , strong topology or standard topology . ${\displaystyle \tau _{w}}$ ${\displaystyle \tau _{0}}$

Sets from the weak topology are marked with the prefix "weak". That's a lot

• weakly terminated when it is the complement of a set in the weak topology.
• weakly compact if there is a finite partial cover for every cover with sets from the weak topology .

Likewise, the weak closure of a set is the smallest weakly closed set that contains. The further naming follows this scheme. ${\displaystyle A}$ ${\displaystyle A}$

See also

• Limited weak - * topology
• Product topology
• Weak - * topology
• Strong operator topology

Web links

• MI Voitsekhovskii: weak topology . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Weak Topology . In: MathWorld (English).

literature

• Winfried Kaballo: Advanced Course Functional Analysis and Operator Theory . Distributions - locally convex methods - spectral theory. 1st edition. Springer-Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-37793-8 , doi : 10.1007 / 978-3-642-37794-5 .
• 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 .
• Manfred Dobrowolski: Applied Functional Analysis . Functional analysis, Sobolev spaces and elliptic differential equations. 2nd, corrected and revised edition. Springer-Verlag, Berlin / Heidelberg 2010, ISBN 978-3-642-15268-9 , doi : 10.1007 / 978-3-642-15269-6 .

Individual evidence

1. ^ Werner: Functional Analysis. 2011, p. 405.
2. ↑ Kaballo: Advanced Course Functional Analysis and Operator Theory. 2014, p. 150.
3. ^ Werner: Functional Analysis. 2011, pp. 410-411.
4. ↑ Kaballo: Advanced Course Functional Analysis and Operator Theory. 2014, p. 175.
5. ↑ Dobrowolski: Applied Functional Analysis. 2010, p. 55.