weak-* topology

Let X be a locally convex topological vector space (over or ), and let X* be the set of continuousMathworldPlanetmathPlanetmath linear functionalsMathworldPlanetmathPlanetmath on X (the continuous dual of X). If fX* then let pf denote the seminormMathworldPlanetmath pf(x)=|f(x)|, and let px(f) denote the seminorm px(f)=|f(x)|. Obviously any normed space is a locally convex topological vector space so X could be a normed space.


The topology on X defined by the seminorms {pffX*} is called the weak topology and the topology on X* defined by the seminorms {pxxX} is called the weak-* topology.

The weak topology on X is usually denoted by σ(X,X*) and the weak-* topology on X* is usually denoted by σ(X*,X). Another common notation is (X,wk) and (X*,wk-*)

Topology defined on a space Y by seminorms pι, ιI means that we take the sets {yYpι(y)<ϵ} for all ιI and ϵ>0 as a subbase for the topology (that is finite intersectionsMathworldPlanetmath of such sets form the basis).

The most striking result about weak-* topology is the Alaoglu’s theorem which asserts that for X being a normed space, a closed ball (in the operator normMathworldPlanetmath) of X* is weak-* compact. There is no similarPlanetmathPlanetmath result for the weak topology on X, unless X is a reflexive space.

Note that X* is sometimes used for the algebraic dual of a space and X is used for the continuous dual. In functional analysisMathworldPlanetmath X* always means the continuous dual and hence the term weak-* topology.


  • 1 John B. Conway. , Springer-Verlag, New York, New York, 1990.
Title weak-* topology
Canonical name WeakTopology
Date of creation 2013-03-22 15:07:05
Last modified on 2013-03-22 15:07:05
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 46A03
Synonym weak-* topology
Synonym weak-* topology
Synonym weak-star topology
Related topic WeakHomotopyAdditionLemma
Defines weak topology