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Inner Product Space

Description

A vector space that possesses an inner product, i.e., a vector space which allows computation of vector length as well as both angle and distance between vectors. Each inner product ⟨·, ·⟩_() defines a norm of the form left double bracketing bar x right double bracketing bar = sqrt(⟨x, x⟩_()) and implicitly, an inner product 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 | normed space | pseudo-metrizable space | quasi-barrelled space | quasi-normed space | seminormed space | topological vector 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)

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. 14, 1966. Lawrence Narici and Edward Beckenstein. Topological Vector Spaces, 2nd ed. p. 17, 2011. Walter Rudin. Functional Analysis, 2nd ed. 1991. Helmut H. Schaefer and Manfred P.H. Wolff. Topological Vector Spaces, 2nd ed. p. 44, 1999. Yau-Chuen Wong. Introductory Theory of Topological Vector Spaces. p. 28, 1992.

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