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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.

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