Bilinear Forms

Abstract

This chapter is devoted to the study of certain types of functions, called bilinear, which are defined for pairs of vectors (x,y′) where x is in a left vector space R andy′ is in a right vector space R′. The values of g(x,y′) are assumed to belong to Δ, and the functions of one variable g _x (y′) = g(x,y′) and g _y′ (x) = g(x,y′) obtained by fixing the other variable are linear. Of particular interest are the non-degenerate bilinear forms. These determine 1–1 linear transformations of R′ onto the space of linear functions on R. Consequently, if A is a linear transformation in R, there is a natural way of associating with it a transposed linear transformation in R′.