important generalizations of inner products, in that often all that is required is that the map be an isomorphism, not positivity. For example, a manifold with an inner product structure on its tangent spaces is a Riemannian manifold, while relaxing this to a symmetric nondegenerate form yields a pseudo-Riemannian manifold. Infinite dimensions Note that in an infinite-dimensional space, we can have a bilinear form ƒ for which is injective but not surjective. For example, on the space of continuous functions on a closed bounded interval, the form is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies for all implies that In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to be weakly nondegenerate. Terminology If f vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form f on V the set of vectors forms a totally degenerate subspace of V. The map f is nondegenerate if and only if this subspace is trivial. Geometrically, an isotropic line of the quadratic form corresponds to a point of the associated quadric hypersurface in projective space. Such a line is additionally isotropic for the bilinear form if and only if the corresponding point is a singularity. Hence, over an algebraically closed field, Hilbert's Nullstellensatz guarantees that the quadratic form always has isotropic lines, while the bilinear form has them if and only if the surface is singular. See also Citations Bilinear forms Functional analysis pl:Forma dwuliniowa#Własności