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kreins

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There are 6 letters in KREINS ( E1I1K5N1R1S1 )

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6 letters out of KREINS

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Kreins might refer to
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).
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