Tensor norms and operators in the category of Banach spaces

Abstract

The theory of dual functors in the category of Banach spaces is applied to the study of tensor norms in the sense of Grothendieck. The dual functors of the tensor norms arising from the projective and inductive tensor product as well as from more general tensor norms, such as the norms dp of Saphar, are identified as various spaces of operators, which include p-integral and absolutely p-summing operators. Properties of these operators are then easily derived by categorical means. Applications of the methods provide simplified proofs of composition theorems and the characterization of dual spaces of type (L).