Weak - * topology

The weak - * - topology is an important topology on the dual space of a normalized (or more generally locally convex ) space. The meaning is based u. a. on the Banach-Alaoglu theorem , according to which the unit sphere in the dual space is compact with respect to this topology . The weak - * - topology plays an important role in many functional analytical constructions, for example in the Gelfand transformation or in the Mackey-Arens theorem , which describes those topologies on a locally convex space that lead to the same topological dual space as the initial topology.

definition

Each element of a normalized or locally convex general - vector space ( here or ) defined by the formula a linear functional on the topological dual space . The weak - * - topology is defined as the weakest topology in which all these pictures ever makes. ${\displaystyle x}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle E}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\hat {x}}(f):=f(x)}$ ${\displaystyle E'\,}$ ${\displaystyle E'\,}$ ${\displaystyle {\hat {x}}:E'\rightarrow \mathbb {K} }$

A more concrete definition can be obtained by specifying an environment base . For make up the quantities ${\displaystyle f\in E'}$

${\displaystyle U_{f}(x_{1},\ldots ,x_{n},\epsilon ):=\{g\in E';|f(x_{j})-g(x_{j})|<\epsilon ,j=1,\ldots ,n\}}$,

where , a neighborhood basis weak - * - open sets of f. The weak - * - topology is often referred to with w ^* , after the English term weak - * - topology, or with , to indicate the origin as the initial topology. ${\displaystyle x_{1},\ldots ,x_{n}\in E,n\in {\mathbb {N} },\epsilon >0}$ ${\displaystyle \sigma (E\,',E)}$

convergence

A sequence in (or more generally a network ) converges in the weak - * topology to if and only if ${\displaystyle (f_{n})_{n\in {\mathbb {N} }}}$ ${\displaystyle E^{\prime }}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle f\in E^{\prime }}$

${\displaystyle \lim _{n\to \infty }f_{n}(x)=f(x)\quad ({\text{bzw.}}\ \lim _{i\in I}f_{i}(x)=f(x))}$

applies to all . That is why the weak - * - topology is also called the topology of point-wise convergence . ${\displaystyle x\in E}$

Semi-norms

The dual space with the weak - * - topology is a locally convex space. The weak - * topology can therefore also be defined by specifying a semi-norm system. With form the semi-norms ${\displaystyle E'}$ ${\displaystyle x_{1},\ldots ,x_{n}\in E,n\in {\mathbb {N} }}$

${\displaystyle p_{x_{1},\ldots ,x_{n}}(f):=\max\{|f(x_{1})|,\ldots ,|f(x_{n})|\}}$,

such a system.

Product topology

It holds because the Cartesian product on the right is nothing but the set of all functions . Since the weak - * - topology, the topology of pointwise convergence is as described above, it is this as a relative topology of the product topology to describe. ${\displaystyle \textstyle E'\subset \prod _{x\in E}\mathbb {K} =\mathbb {K} ^{E}}$ ${\displaystyle E\rightarrow \mathbb {K} }$ ${\displaystyle \textstyle \prod _{x\in E}{\mathbb {K} }}$

According to Tychonoff's theorem, there is a compact subset for every choice of positive real numbers in the product space . This fact is an essential step in the proof of the Banach-Alaoglu Theorem . ${\displaystyle \textstyle \prod _{x\in E}\{\lambda \in {\mathbb {K} };\,|\lambda |\leq r_{\lambda }\}}$ ${\displaystyle r_{\lambda }}$

properties

• The weak - * topology makes it a locally convex space . If one forms the strong dual space with regard to this topology , one obtains , or in short ${\displaystyle \sigma (E',E)}$ ${\displaystyle E'}$ ${\displaystyle \{{\hat {x}};x\in E\}\cong E}$

${\displaystyle (E',\sigma (E',E))'\cong E}$.

• Probably the most important property in the case of normalized spaces is treated in the Banach-Alaoglu theorem , that is the weak - * - compactness of the unit sphere in the dual space.
• The canonical embedding of a Banach space in its bidual space can be viewed as a subspace of . The Hahn-Banach theorem shows that with respect to the weak - * - topology densely located. With the help of the separation theorem , one can show that this tightness relationship is also correct for standardized spaces for the standard spheres , that is, the one that goes back to Herman H. Goldstine applies ${\displaystyle E}$ ${\displaystyle E''}$ ${\displaystyle E}$ ${\displaystyle E''}$ ${\displaystyle E}$ ${\displaystyle \sigma (E'',E')}$

Goldstine's sentence : lies tightly in . ${\displaystyle \{x\in E;\|x\|\leq 1\}}$ ${\displaystyle \sigma (E'',E')}$ ${\displaystyle \{x''\in E'';\|x''\|\leq 1\}}$

See also

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

literature

• Klaus Floret, Joseph Wloka: Introduction to the theory of locally convex spaces (= Lecture Notes in Mathematics. Vol. 56, ISSN  0075-8434 ). Springer, Berlin et al. 1968.