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Locally Convex


A topology tau on a topological vector space X=(X,tau) (with X usually assumed to be T2) is said to be locally convex if tau admits a local base at 0 consisting of balanced, convex, and absorbing sets. In some older literature, the definition of locally convex is often stated without requiring that the local base be balanced or absorbing.

It is not unusual to blur the distinction as to whether "locally convex" applies to the topology tau on X or to X itself.

The above definition can also be stated in terms of seminorms. In particular, a topological vector space (X,tau) (with X assumed T^2) is locally convex if tau is generated by a family P of seminorms satisfying

  intersection _(p in P){x in X:p(x)=0}=0

where 0 denotes the zero vector in X and is different from 0 which denotes the element 0 in the scalar field of X. The condition (1) above ensures that X is T^2; removal of this criterion on X allows one to remove condition (1), whereby (X,tau) is locally convex if and only if tau is generated by a family P of seminorms.

The seminorm condition illustrates why local convexity is a desirable property. In particular, topological vector spaces which are locally convex can be thought of as generalizations of normed spaces, thereby allowing considerable functional analysis to be done even without the existence of a norm.


See also

Convex Set, Normed Space, Seminorm

This entry contributed by Christopher Stover

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References

Wong, Y. Introductory Theory of Topological Vector Spaces. New York: Dekker, 1992.Conway, J. A Course in Functional Analysis. New York: Springer-Verlag, 1990.Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Cite this as:

Stover, Christopher. "Locally Convex." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LocallyConvex.html

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