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 majorant topology. Krein spaces are named in honor of the Ukrainian mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).
Read more about Indefinite Inner Product Space: Inner Products and The Metric Operator, Properties and Applications, Isotropic Part and Degenerate Subspaces, Pontrjagin Space, Pesonen Operator
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