Indefinite inner product space

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.

Source: Wikipedia — Indefinite inner product space (CC BY-SA 4.0)

Indefinite inner product space

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.

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Source: Wikipedia "Indefinite inner product space" · CC BY-SA 4.0

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