ABSTRACT
This chapter is devoted to bilinear maps and forms. We previously defined the concept of an m-linear map from vector spaces V1, . . . , Vm to the vector space W. A particular important special case is when m = 2 and such functions were referred to as bilinear maps. Bilinear maps are important, in particular, because of their role in the definition of the tensor product of two spaces, which is the subject of the next chapter. Bilinear forms (bilinear maps to F, the underlying field) arise throughout mathematics, in fields ranging from differential geometry and mathematical physics on the one hand to group theory and number theory on the other. In the introductory section of this chapter, we develop some basic properties of bilinear maps and forms, introduce the notion of a reflexive form, and prove that any reflexive form is either alternating or symmetric. The second section is devoted to the structure of symplectic space, a vector space equipped with an alternate form. In the third section, we define the notion of a quadratic form and develop the general theory of an orthogonal space. The final section deals specifically with real orthogonal spaces.
