The theory of the free fermion field is essentially distinct from but nevertheless parallel to that of the free boson field, and the underlying formal analogy is close and useful.

Definition. An*orthogonal space*is a pair (L,*S*) consisting of a real topological vector space L and a given continuous symmetric nondegenerate bilinear form*S*on L.

Concerning the notions of nondegeneracy, symmetry, and of the algebraic topology on a real vector space, see Chapter 1.

Example 2.1. Let M be finite-dimensional real vector space and M^{*}its dual. Let L denote the direct sum M⊕M^{*}, and let*S*