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
Famous quotes containing the words indefinite, product and/or space:
“As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
—Henry David Thoreau (1817–1862)
“The writer’s language is to some degree the product of his own action; he is both the historian and the agent of his own language.”
—Paul De Man (1919–1983)
“Mere human beings can’t afford to be fanatical about anything.... Not even about justice or loyalty. The fanatic for justice ends by murdering a million helpless people to clear a space for his law-courts. If we are to survive on this planet, there must be compromises.”
—Storm Jameson (1891–1986)