Editing Inner product space (section)

===Nondegenerate conjugate symmetric forms===
{{Main|Pseudo-Euclidean space}}
Alternatively, one may require that the pairing be a [[nondegenerate form]], meaning that for all non-zero <math>x \neq 0</math> there exists some <math>y</math> such that <math>\langle x, y \rangle \neq 0,</math> though <math>y</math> need not equal <math>x</math>; in other words, the induced map to the dual space <math>V \to V^*</math> is injective. This generalization is important in [[differential geometry]]: a manifold whose tangent spaces have an inner product is a [[Riemannian manifold]], while if this is related to nondegenerate conjugate symmetric form the manifold is a [[pseudo-Riemannian manifold]]. By [[Sylvester's law of inertia]], just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with {{em|nonzero}} weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in [[Minkowski space]] is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four [[Dimension (mathematics)|dimensions]] and indices 3 and 1 (assignment of [[Sign (mathematics)|"+" and "−"]] to them [[Sign convention#Metric signature|differs depending on conventions]]).

Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism <math>V \to V^*</math>) and thus hold more generally.