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Hilbert Space

Description

A Hilbert space is a type of abstract vector space that generalizes the notion of Euclidean space. More precisely, a Hilbert space is defined to be a vector space which possesses the structure of an inner product (thus allowing length and angle to be measured) and for which the metric induced by the inner product is sequentially complete.

Relationship graph

Relationship graph

More general classifications

Baire space | Banach space | barrelled space | bornological space | compactly generated space | complete space | convenient space | Fréchet space | F-space | inner product space | locally complete space | locally convex space | Mackey space | metrizable space | normed space | pseudo-complete space | pseudo-metrizable space | quasi-Banach space | quasi-barrelled space | quasi-complete space | quasi-normed space | reflexive space | seminormed space | semi-reflexive space | sequentially complete space | stereotype space | topological vector space | webbed space

Examples

A^2(D, dλ^2) | a^2(D, dλ^2) | h^2 | H^2 | L^2(D, dλ^2) | ℓ^2(Z^+, dη)

History

David Hilbert (mathematician)

Erhard Schmidt | Frigyes Riesz

Timeline

Timeline

References

Gustave Choquet. Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces. p. 26, 1969. John Horvath. Topological Vector Spaces and Distributions. Vol. I. p. 15, 1966. Gottfried Köthe. Topological Vector Spaces. I. p. 23, 1969. Lawrence Narici and Edward Beckenstein. Topological Vector Spaces, 2nd ed. p. 18, 2011. Helmut H. Schaefer and Manfred P.H. Wolff. Topological Vector Spaces, 2nd ed. p. 44, 1999. François Trèves. Topological Vector Spaces, Distributions and Kernels. p. 3115, 1967. Albert Wilansky. Modern Methods in Topological Vector Spaces. p. 169, 1978. Yau-Chuen Wong. Introductory Theory of Topological Vector Spaces. p. 30, 1992.

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