The Projective Tensor Product

Abstract

In this chapter we investigate the simplest way to norm the tensor product of two Banach spaces. The projective tensor product linearizes bounded bilinear mappings just as the algebraic tensor product linearizes bilinear mappings. The projective tensor product derives its name from the fact that it behaves well with respect to quotient space constructions. The projective tensor product of ℓ₁ with X gives a representation of the space of absolutely summable sequences in X and projective tensor products with L(µ)lead to a study of the Bochner integral for Banach space valued functions. We also introduce the class of ℒ-spaces, whose finite dimensional structure is like that of ℓ₁. We study some techniques that make use of the Rademacher functions, including the Khinchine inequality. Finally, interpreting the elements of a projective tensor product as bilinear forms or operators leads to the introduction of the concept of nuclearity.