On Some Duality Relations in the Theory of Tensor Products

We study several classical duality results in the theory of tensor products, due mostly to Grothendieck, providing new proofs as well as new results. In particular, we show that the canonical mapping Y ∗ ⊗π X → (L(X, Y ), τ)∗ is not always injective, answering a problem of Defant and Floret. We use the machinery of vector measures to give new proofs of the dualitites (X⊗ε Y )∗ = N (X, Y ∗), whenever Y ∗ has the RNP, and (a slight improvement of) the result of Rosenthal (X ⊗ε Y )∗ ⊂ F(X, Y ∗), whenever `1 6↪→ Y . 1. Inroduction and preliminaries The goal of the present note is to study several classical duality results in the theory of tensor products, due mostly to Grothendieck, providing new proofs as well as new results. An important result in the topological theory of tensor products is the theorem of Grothendieck that gives a description the linear topological dual of the space of bounded linear operators L(X, Y ) equipped with the τ -topology of uniform convergence on compact sets. According to this result the continuous linear functionals on (L(X, Y ), τ) consist of all