In mathematics, in the field of functional analysis, an indefinite inner product space {\displaystyle } 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 = d e f ⟨ x, J y ⟩, {\displaystyle \ {\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. Wikipedia