Indefinite inner product space  In mathematics, in the field of functional analysis, an indefinite inner product space   is an infinite-dimensional complex vector space formula_2 equipped with both an indefinite inner product   and a positive semi-definite inner product   where the metric operator formula_5 is an endomorphism of formula_2 obeying   The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on formula_2 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 formula_5"-space") if formula_10 is positive definite and formula_2 possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989). Consider a complex vector space formula_2 equipped with an indefinite hermitian form formula_13. In the theory of Krein spaces it is common to call such an hermitian form an indefinite inner product. The following subsets are defined in terms of the square norm induced by the indefinite inner product:  A subspace formula_19 lying within formula_20 is called a "neutral subspace". Similarly, a subspace lying within formula_21 (formula_22) is called "positive" ("negative") "semi-definite", and a subspace lying within formula_23 (formula_24) is called "positive" ("negative") "definite". A subspace in any of the above categories may be called "semi-definite", and any subspace that is not semi-definite is called "indefinite". Let our indefinite inner product space also be equipped with a decomposition into a pair of subspaces formula_25, called the "fundamental decomposition", which respects the complex structure on formula_2. Hence the corresponding linear projection operators formula_27 coincide with the identity on formula_28 and annihilate formula_29, and they commute with multiplication by the formula_30 of the complex structure. If this decomposition is such that formula_31 and formula_32, then formula_2 is called an indefinite inner product space; if formula_34, then formula_2 is called a Krein space, subject to the existence of a majorant topology on formula_2 (a locally convex topology where the inner product is jointly continuous).