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Locally Convex Space

Description

A locally convex space is a topological vector space which has a local base made up of convex neighborhoods of the vector 0 where here, a local base about a vector x in X is a collection of neighborhoods B of x satisfying the condition that any other neighborhood U of x contains an element of B.

Relationship graph

Relationship graph

More general classifications

topological vector space

Examples

A^1(D, dλ^2) | A^2(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^2(D, dλ^2) | L^∞(D, dλ^2) | ℬ_0(D, dλ^2) | c_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. p. 7, 1978. Gustave Choquet. Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces. pp. 333 and 351, 1969. Alexander Grothendieck. Topological Vector Spaces. p. 22, 1973. John Horvath. Topological Vector Spaces and Distributions. Vol. I. p. 86, 1966. Taqdir Husain. The Open Mapping and Closed Graph Theorems in Topological Vector Spaces. p. 16, 1965. Taqdir Husain and S.M. Khaleelulla. Barrelledness in Topological and Ordered Vector Spaces. p. 8, 1978. Gottfried Köthe. Topological Vector Spaces. I. p. 202, 1969. Lawrence Narici and Edward Beckenstein. Topological Vector Spaces, 2nd ed. p. 82, 2011. Alexander Provan Robertson and Wendy Robertson. Topological Vector Spaces. p. 10, 1964. Helmut H. Schaefer and Manfred P.H. Wolff. Topological Vector Spaces, 2nd ed. p. 47, 1999. François Trèves. Topological Vector Spaces, Distributions and Kernels. p. 58, 1967. Albert Wilansky. Modern Methods in Topological Vector Spaces. pp. 51, 54, 91, and 98, 1978. Yau-Chuen Wong. Introductory Theory of Topological Vector Spaces. p. 113, 1992.

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