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In mathematics, in the field of functional analysis, an Indefinite inner product space * * * * ( * K * , * ⟨ * ⋅ * , * * ⋅ * ⟩ * , * J * ) * * * {\displaystyle (K,\langle \cdot ,\,\cdot \rangle ,J)} * is an infinite-dimensional complex vector space * * * * K * * * {\displaystyle K} * equipped with both an indefinite inner product * * * * * ⟨ * ⋅ * , * * ⋅ * ⟩ * * * * {\displaystyle \langle \cdot ,\,\cdot \rangle \,} * and a positive semi-definite inner product * * * * * ( * x * , * * y * ) * * * * * * = * * * * d * e * f * * * * * * * ⟨ * x * , * * J * y * ⟩ * , * * * {\displaystyle (x,\,y)\ {\stackrel {\mathrm {def} }{=}}\ \langle x,\,Jy\rangle ,} * where the metric operator * * * * J * * * {\displaystyle J} * is an endomorphism of * * * * K * * * {\displaystyle K} * obeying * * * * * * J * * 3 * * * = * J * . * * * * {\displaystyle J^{3}=J.\,} * The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on * * * * K * * * {\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. * An indefinite inner product space is called a Krein space (or * * * * J * * * {\displaystyle J} * -space) if * * * * ( * x * , * * y * ) * * * {\displaystyle (x,\,y)} * is positive definite and * * * * K * * * {\displaystyle K} * possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989). |