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Topological Vector Space


A vector space with a T2-space topology such that the operations of vector addition and scalar multiplication are continuous. The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a Hilbert space and a Banach space are topological vector spaces.

The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the Banach space L^1(X), one of the L-p-spaces, is used. But if one is interested in pointwise convergence, then no norm will suffice. Instead, for each x in X define the seminorm

 ||f||_x=|f(x)|

on the vector space of functions on X. The seminorms define a topology, the smallest one in which the seminorms are continuous. So limf_n=f is equivalent to limf_n(x)=f(x) for all x in X, i.e., pointwise convergence. In a similar way, it is possible to define a topology for which "convergence" means uniform convergence on compact sets.


See also

Banach Space, Hilbert Space, Seminorm, Topological Space, Vector Space

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References

Köthe, G. Topological Vector Spaces. New York: Springer-Verlag, 1979.Zimmer, R. Essential Results in Functional Analysis. Chicago: University of Chicago Press, pp. 13-17, 1990.

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Topological Vector Space

Cite this as:

Weisstein, Eric W. "Topological Vector Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TopologicalVectorSpace.html

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