Quasinormable Spaces and the Problem of Topologies of Grothendieck

This article is dedicated to the study of quasinormable injective tensor products of locally convex spaces and quasinormable spaces of continuous linear operators. The stability of the quasinormability is obtained in the frame of the class of spaces which are quasinormable by operators; this class, introduced and studied here, contains many function spaces. The problems considered in the article are closely related to the problem of topologies of Grothendieck. A characterization of the quasinormable spaces which are (FBa)-spaces in the sense of Taskinen is obtained and new examples and counterexamples are given. In particular we show that the quasinormable space l p+ is a concrete example of a non-(FBa)-space. Grothendieck (see 30, 31]) studied locally convex properties of function spaces, such as spaces of sequences, of diierentiable functions, analytic functions, distributions, etc. There are a lot of examples of spaces of vector-valued functions which can be represented as tensor products or as spaces of continuous and linear mappings and it is convenient to know their topological structure. The aim of this article is to study the stability of the property of being quasi-normable under the formation of injective tensor products or of spaces of continuous and linear mappings. The class of quasinormable locally convex spaces was introduced and studied by Grothendieck as a class containing most of the usual function spaces. Banach spaces and Schwartz spaces are examples of quasi-normable spaces. The typical examples of quasinormable spaces which are neither normable nor Montel are the following: the space C(X) endowed with the compact open topology for every completely regular Hausdorr space X , the spaces C k (() (k 2 N, an open subset of R N), every non-trivial quojection, every non-trivial (gDF)-space which is not Montel, the spaces B 0 (R N) and B(R N) of Schwartz and the local spaces B loc p;k (() of HH ormander. More precisely we investigate the following questions: