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Normed Space

Description

A vector space X that possesses a norm, i.e., a vector space endowed with a form left double bracketing bar x right double bracketing bar which separates points and is absolutely homogeneous and subadditive. These three properties are called the norm axioms and they imply that a normed space allows meaningful computation of vector length and distance between vectors. Every norm defines a topology and implicitly, each normed space is equipped with the topology determined by its norm.

Relationship graph

Relationship graph

More general classifications

bornological space | compactly generated space | locally convex space | Mackey space | metrizable space | pseudo-metrizable space | quasi-barrelled space | quasi-normed space | seminormed space | 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

Gustave Choquet. Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces. p. 25, 1969. Alexander Grothendieck. Topological Vector Spaces. p. 18, 1973. John Horvath. Topological Vector Spaces and Distributions. Vol. I. pp. 6 and 10, 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. 9, 1978. Gottfried Köthe. Topological Vector Spaces. I. p. 123, 1969. Lawrence Narici and Edward Beckenstein. Topological Vector Spaces, 2nd ed. p. 14, 2011. Walter Rudin. Functional Analysis, 2nd ed. 1991. Helmut H. Schaefer and Manfred P.H. Wolff. Topological Vector Spaces, 2nd ed. p. 41, 1999. François Trèves. Topological Vector Spaces, Distributions and Kernels. p. 95, 1967. Albert Wilansky. Modern Methods in Topological Vector Spaces. p. 18, 1978. Yau-Chuen Wong. Introductory Theory of Topological Vector Spaces. p. 13, 1992.

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