Pontryagin space

In mathematics, in the field of functional analysis, an indefinite inner product space

is an infinite-dimensional complex vector space ${\displaystyle K}$ equipped with both an indefinite inner product

and a positive semi-definite inner product

where the metric operator ${\displaystyle J}$ is an endomorphism of ${\displaystyle K}$ obeying

The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on ${\displaystyle K}$ 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.

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