Indefinite metric

A term used in the theory of spaces with an indefinite metric (cf. Space with an indefinite metric) for denoting (depending on the type of space) either a bilinear form, a sesquilinear form or a (non-linear) functional of a certain degree of homogeneity, defined on the space under consideration. An indefinite metric is not a metric (that is, a distance), and the epithet "indefinite" means that either the sesquilinear form is not positive definite or the functional is not a power of a norm on the space. The various types of indefinite metrics are called $G$-metric, $I$-metric, $J$-metric (see Hilbert space with an indefinite metric; Pontryagin space).