Weak topology#The weak-.2A topology

The weak -* topology is an important topology on the dual space of a normed (or more generally locally convex ) space. The significance is based inter alia on the set of Banach Alaoglu, after which the unit ball is compact in the dual space with respect to this topology. The weak -* topology plays an important role in many functional analytic structures, such as in the Gelfand transform or in the set of Mackey - Arens, which describes those topologies on a locally convex space, lead to the same topological dual space as the original topology.

Definition

Each element of a normalized or general locally convex 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 that makes all these pictures ever.

A slightly more specific definition is obtained by specifying a neighborhood basis. For forming the quantities

Whereby, an environment base weak - * - open sets of f, the weak -* topology is often referred to with or.

Convergence

A sequence in (or more generally a network ) converges to in the weak -* topology if

Applies to all. Therefore, we call the weak -* topology, the topology of pointwise convergence.

Seminorms

The dual space is the weak -* topology of a locally convex space. The weak -* topology can therefore also be defined by specifying a semi-norms system. With form the semi-norms

Such a system.

Properties

• The weak -* topology makes it a locally convex space. If one forms with respect to this topology, the strong dual space, one obtains, or shortly

• The most important feature in the case of normed spaces is treated in the Banach - Alaoglu, which is the weak -* compactness of the unit ball in the dual space.
• By the canonical embedding of a Banach space in its Bidualraum can be considered as subspace of. The Hahn- Banach theorem shows that tight with respect to the weak -* topology is. Using the separation theorem, one can show that this tightness relationship is also true for the unit balls in normed spaces, ie, it applies the receding on Herman H. Goldstine