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Ebook Advanced calculus: Part 2

(BQ) Part 2 book "Advanced calculus" has contents: Multilinear functionals, integration, differentiable manifolds, the integral calculus on manifolds, exterior calculus, potential theory in E^n, classical mechanics. | CHAPTER 7 MULTILINEAR FUNCTIONALS This chapter is principally for reference. Although most of the proofs will be included, the reader is not expected to study them. Our goal is a collection of basic theorems about alternating multilinear functionals, or exterior forms, and the determinant function is one of our rewards. 1. BILINEAR FUNCTIONALS We have already studied various aspects of bilinear functionals. We looked at their duality implications in Section 6, Chapter 1, we considered the "canonical forms" of symmetric bilinear functionals and their equivalent quadratic forms in Section 7, Chapter 2, and, of course, the whole scalar product theory of Chapter 5 is the theory of a still more special kind of bilinear functional. In this chapter we shall restrict ourselves to bilinear and multilinear functionals over finite-dimensional spaces, and our concerns are purely algebraic. We begin with some material related to our earlier algebra. If V and Ware finite-dimensional vector spaces, then the set of all bilinear functionals on V X W is pretty clearly a vector space. We designate it V* ® W* and call it the tensor product of V* and W*. Our first theorem simply states something that was implicit in Theorem 6.1 of Chapter 1. TheoreIn 1.1. The vector spaces V* ® W*, Hom(V, W*), and Hom(W, V*) are naturally isomorphic. Proof. We saw in Theorem 6.1 of Chapter 1 that eachfin V* ® W* determines a linear mapping a 1--+ fa from W to V*, wherefa(t) = f(t, a), and we also noted that this correspondence from V* ® W* to Hom(W, V*) is bijective. All that the present theorem adds is that this bijective correspondence is linear and so constitutes a natural isomorphism, as does the similar one from V* ® W* to Hom(V, W*). To see this, leth be the bilinear functional corresponding to Tin Hom(V, W*). Then f(T+S) = h fs, for f(T+S)(a, (3) = (T S)(a))({3) = (T(a) S(a))({3) = (T(a))({3) (S(a))({3) = h(a, (3) fs(a, (3). We can do the same for homogeneity. The isomorphism of V* ® W* with