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:
“There are times when they seem so small! And then again, although they never seem large, there is a vastness behind them, a past of indefinite complexity and marvel, an amazing power of absorbing and assimilating, which forces one to suspect some power in the race so different from our own that one cannot understand that power. And ... whatever doubts or vexations one has in Japan, it is only necessary to ask oneself: “Well, who are the best people to live with?””
—Lafcadio Hearn (1850–1904)
“These facts have always suggested to man the sublime creed that the world is not the product of manifold power, but of one will, of one mind; and that one mind is everywhere active, in each ray of the star, in each wavelet of the pool; and whatever opposes that will is everywhere balked and baffled, because things are made so, and not otherwise.”
—Ralph Waldo Emerson (1803–1882)
“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, ‘lelo No’Eau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)