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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 An indefinite inner product space is called a Krein space (or -space) if is positive definite and possesses a . Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).