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Weak-* Topology


Let X=(X,tau) be a topological vector space whose continuous dual X^* may or may not separate points (i.e., may or may not be T2). The weak-* (pronounced "weak star") topology on X^* is defined to be the X-topology on X^*, i.e., the coarsest topology (the topology with the fewest open sets) under which every element x in X corresponds to a continuous map on X^*. The weak-* topology is sometimes called the ultraweak topology or the sigma-weak topology.

The fundamental observation to the above-stated definition is that every element x in X induces a linear functional f_(x) on X^*. In particular, f_(x) of the form

 f_(x)Lambda=Lambda(x)

for every element Lambda in X^*, and because of this, one can view the space X as a collection of linear functionals on X^* and hence can define the X-topology thereon.

Immediately following from the above is the fact that the weak-* topology is a special case of a more general concept. In particular, given a nonempty family Gamma of mappings from a set X to a topological space Y, one can define a topology tau_Gamma to be the collection of all unions and finite intersections of sets of the form f^(-1)(V) with f in Gamma and V an open set in Y. The topology tau_Gamma is often called the Gamma-topology on X and/or the weak topology on X induced by Gamma, whereby it follows that the above-stated definition corresponds to the special case of Gamma=X for X a topological vector space.

The weak-* topology is fundamental throughout functional analysis, playing a fundamental role in a number of important theorems including the Banach-Alaoglu theorem.


See also

Banach-Alaoglu Theorem, Topological Vector Space, Weak Topology

This entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Cite this as:

Stover, Christopher. "Weak-* Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Weak-StarTopology.html

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