On homomorphisms between locally convex spaces

Abstract

Homomorphisms $f : E → F$ between locally convex spaces $E, F$, i.e. continuous linear maps which are open onto the range, occur quite often and they are nice to handle.Unfortunately, the stability properties of the class of homomorphisms are poor.For instance, a homomorphism $f : E → F$ will in general not remain a homomorphism, if E and F are endowed with, for instance, their strong topology; the transpose $f^t : F' → E'$ will usually not be a homomorphism, and the behaviour of the bitranspose is still worse. The investigation of homomorphisms has a good tradition,in fact, it goes back to Banach and was dealt with afterwards by Dieudonné, L. Schwartz, Grothendieck and Köthe (see for example [12]). The purpose of this article is twofold: first, to study the stability behaviour of the class of homomorphisms with a bit of a systematic touch (see (1.4),(1.8), (2.3), (2.5)); and second, to apply new methods and results from the recent development of the structure theory of Fréchet, $LB$- and $LF$-spaces to the context of homomorphisms. For instance, we obtain a(nother) characterization for the quasinormability of Fréchet spaces E by the property that for every monomorphism $j: E→ F$ with F Fréchet, j remains a homomorphism for the topology of uniform convergence on strongly compact sets both on E and on F (see (1.9), (1.10)).Proposition (2.7) presents a general background for the fact that for the famous quotient map $g : E → \ell¹$ with E Fréchet Montel, the transpose is not a monomorphism for the weak (sic!) topologies. In section 3, where we deal with the bitranspose of homomorphisms, we give an example of a quotient map $q : E → F$ with E Fréchet such that $q^tt : E'' + F''$ is not a homomorphism between the strong biduals. Finally, we present a fairly general condition on a strict $LF$-space, under which its strong bidual will again be an $LF$-space.