Supersymmetric Hilbert space.

A generalization is given of the notion of a symmetric bilinear form over a vector space, which includes variables of positive and negative signature ("supersymmetric variables"). It is shown that this structure is substantially isomorphic to the exterior algebra of a vector space. A supersymmetric extension of the second fundamental theorem of invariant theory is obtained as a corollary. The main technique is a supersymmetric extension of the standard basis theorem. As a byproduct, it is shown that supersymmetric Hilbert space and supersymplectic space are in natural duality.