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Weak Topology


Let X=(X,tau) be a topological vector space whose continuous dual X^* separates points (i.e., is T2). The weak topology tau_w on X is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of X^* remains continuous on X. To differentiate the topologies tau and tau_w, tau is sometimes referred to as the strong topology on X.

Note 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-often called the Gamma-topology on X and/or the weak topology on X induced by Gamma-is the coarsest topology in which every element f in Gamma is continuous on X and so it follows that the above-stated definition corresponds to the special case of Gamma=X^* for X a topological vector space.


See also

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/WeakTopology.html

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