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Topological Vector Space

Description

A topological vector space is a vector space X over a (topological) field F (typically assumed to be the fields R or C of real or complex numbers, respectively) that is endowed with a topology τ such that both vector addition X×X->X and scalar multiplication F×X->X is τ-continuous. Every topological vector space X is trivially an abelian topological group, and also has a continuous dual space X^* consisting of all continuous linear maps X->F from X to the base field F. Note that while some authors insist that the topology τ be Hausdorff, this restriction is not mathematically necessary. Topological vector spaces are of interest in a number of fields including functional analysis and a number of well-studied classes of spaces (e.g., Banach spaces and Hilbert spaces) are topological vector spaces.

Relationship graph

Relationship graph

Examples

A^1(D, dλ^2) | A^2(D, dλ^2) | A^-∞(D, dλ^2) | ℬ(D, dλ^2) | L^∞(T;X) | a^1(D, dλ^2) | a^2(D, dλ^2) | ℬ^h(D, dλ^2) | h^2 | h^∞ | ℬ_0^h(D, dλ^2) | H^2 | H^∞ | L^0(D, dλ^2) | L^2(D, dλ^2) | L^∞(D, dλ^2) | ℬ_0(D, dλ^2) | c_0(Z^+, dη) | ℓ^0(Z^+, dη) | ℓ^2(Z^+, dη) | ℓ^∞(Z^+, dη)

References

Norbert Adasch, Bruno Ernst, and Dieter Keim. Topological Vector Spaces. The Theory without Convexity Conditions. 1978. Sergei Akbarov. "Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra." Journal of Mathematical Sciences 113, 179-349, 2003. Gustave Choquet. Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces. 1969. Alexander Grothendieck. Topological Vector Spaces. 1973. John Horvath. Topological Vector Spaces and Distributions. Vol. I. 1966. Taqdir Husain. The Open Mapping and Closed Graph Theorems in Topological Vector Spaces. 1965. Gottfried Köthe. Topological Vector Spaces. I. 1969. Gottfried Köthe. Topological Vector Spaces. II. 1979. Lawrence Narici and Edward Beckenstein. Topological Vector Spaces, 2nd ed. 2011. Alexander Provan Robertson and Wendy Robertson. Topological Vector Spaces. 1964. Helmut H. Schaefer. Topological Vector Spaces. 1966. Helmut H. Schaefer and Manfred P.H. Wolff. Topological Vector Spaces, 2nd ed. 1999. François Trèves. Topological Vector Spaces, Distributions and Kernels. 1967. Albert Wilansky. Modern Methods in Topological Vector Spaces. 1978. Yau-Chuen Wong. Introductory Theory of Topological Vector Spaces. 1992.

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