Topological vector spaces | Operator theory

Indefinite inner product space

In mathematics, in the field of functional analysis, an indefinite inner product space is an infinite-dimensional complex vector space equipped with both an indefinite inner product and a positive semi-definite inner product where the metric operator is an endomorphism of obeying The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space. An indefinite inner product space is called a Krein space (or -space) if is positive definite and possesses a . Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989). (Wikipedia).

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Inner products (video 3): Definition

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From playlist Inner Products

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Math 060 Fall 2017 110817C Inner Product Spaces 1

Definition of inner product space. Examples. Definitions: orthogonal, norm, vector projection, scalar projection. Pythagorean theorem (in inner product space).

From playlist Course 4: Linear Algebra (Fall 2017)

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Inner Products (video 4): Lengths and Distances, Part 1/2

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

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Inner Products (video 7): Unconventional Inner Products

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From playlist Inner Products

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Inner products (video 8): Outro

Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning

From playlist Inner Products

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Rayleigh Quotient Based Numerical Methods for Eigenvalue Problems Lecture 5

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From playlist Gene Golub SIAM Summer School Videos

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From playlist Orthogonality

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From playlist Mathematics

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