Let be a locally convex topological vector space (over or ), and let be the set of continuous linear functionals on (the continuous dual of ). If then let denote the seminorm , and let denote the seminorm . Obviously any normed space is a locally convex topological vector space so could be a normed space.
The topology on defined by the seminorms is called the weak topology and the topology on defined by the seminorms is called the weak- topology.
The weak topology on is usually denoted by and the weak- topology on is usually denoted by . Another common notation is and
Topology defined on a space by seminorms , means that we take the sets for all and as a subbase for the topology (that is finite intersections of such sets form the basis).
The most striking result about weak- topology is the Alaoglu’s theorem which asserts that for being a normed space, a closed ball (in the operator norm) of is weak- compact. There is no similar result for the weak topology on , unless is a reflexive space.
Note that is sometimes used for the algebraic dual of a space and is used for the continuous dual. In functional analysis always means the continuous dual and hence the term weak- topology.
- 1 John B. Conway. , Springer-Verlag, New York, New York, 1990.
|Date of creation||2013-03-22 15:07:05|
|Last modified on||2013-03-22 15:07:05|
|Last modified by||jirka (4157)|