Weak topology

The term weak topology called in mathematics a special topology on normed vector spaces and is one of the most important concepts of functional analysis. A convergent in the weak topology episode is called weakly convergent.

• 2.1 Definition
• 2.2 Features

The weak topology on a normed vector space

Definition

If a normed vector space and its topological dual space, it is called the initial topology with respect to the weak topology on, as the name is common.

The weak topology is thus the coarsest topology with respect to the norm all continuous linear functionals are continuous. Another formulation is that the inverse images of open sets the standard continuous linear functionals form a sub-base of the weak topology. Specifically, this means that one constructs the open sets as follows:

• Image of all archetypes for and or open,

• Make all the finite intersections of the sets from the first step,

• Make all the associations (even infinite) of sets from the second step.

Properties

• Weak topology is coarser than the topology induced by the standard (standard topology). Standard and weak topology coincide exactly when is finite.
• The closed unit ball of is if and only weakly compact if a reflexive Banach space is.
• Provides you with the weak topology, then remain constant addition and scalar multiplication shortcuts, and is a locally convex space.
• In locally convex topological vector spaces closed and convex subsets are weakly closed.
• The set of Eberlein - Šmulian notes the equivalence of compactness and sequential compactness with respect to the weak topology on Banach spaces.

Definition

A sequence in a normed vector space is called weakly convergent if it converges with respect to the weak topology on.

Properties

• The sequence converges weakly to, applies when in or for all continuous linear functionals ( the topological dual space ).

• There is a Hausdorff space with respect to the weak topology, the weak limit, if it exists at all, clearly.

• Each strongly convergent sequence ( in the norm convergent ) is also weakly convergent, the converse is not true in general. A counter-example is in the following space with the sequence, applies. Indeed, if, then there exists a sequence, so that applies to everyone. This applies to, ie, the sequence against weak. But for all, the result does not converge strongly to 0 Because of the described ambiguity is even considered that no subsequence can converge from strong.

• Each weak convergent sequence of a normed vector space is limited. This property is a consequence of the theorem of Banach - Steinhaus.

• In a reflexive space, each bounded sequence a weakly convergent subsequence. Since every Hilbert space is reflexive, so always has a bounded sequence in a Hilbert space, a weakly convergent subsequence.

• In a Hilbert space is the weak convergence is equivalent to the component-wise convergence with respect to an orthogonal basis.

• By the theorem of Mazur every weakly convergent sequence has in normed vector spaces a sequence of convex combinations of normkonvergente followers.

Rule of thumb for the term " weak" in the Functional Analysis

" Weak" means as much as " with respect to of all continuous linear forms ... ".

Examples:

• One consequence is if and only weakly convergent if it converges with respect to all continuous linear forms.
• A lot is if and only weakly sequentially compact if every sequence from a respect of all continuous linear forms has a convergent subsequence.
• A set is bounded if and only weakly, if at any of the proposed continuous linear forms of the associated image set is a limited amount in the underlying field. Note: Due to the set of Banach - Steinhaus every weakly bounded set is always norm-bounded.

Related terms

• * Weakly Topology: This is the initial topology, which is generated by the elements of when they are identified in functional. She is still coarser than the weak topology. An important statement about the weak -* topology is the set of Banach Alaoglu.