Nuclear Maps and Spaces

Abstract

This final chapter is devoted to the study of a hierarchy of finiteness conditions one can impose on a continuous linear map between two locally convex K-vector spaces V and W. In increasing speciality these are the conditions of being completely continuous, compact, nuclear and of finite rank. The whole theory is intimately connected to the properties of the projective tensor product. We therefore begin in §17 by introducing the inductive and projective tensor product topologies. From §18 on it again will be a standing assumption that the field K is spherically complete. The completely continuous maps are the maps which lie in the closure of the finite rank operators with respect to the topology of bounded convergence. Compact maps were already defined in §16. Any compact map is completely continuous; the converse holds for maps between Banach spaces. Finally, continuous linear maps which factorize through a compact map between Banach spaces are called nuclear. The basic properties of these classes of maps are presented in §§18 and 20. In between in §19 we discuss nuclear spaces. In the second half of §20 these concepts are applied to the study of the continuous linear dual of a projective tensor product.