On homomorphisms between locally convex spaces


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<sup>t</sup> : 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 → \ell<sup>1</sup> 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<sup>tt</sup> : 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.

DOI Code: 10.1285/i15900932v12p27

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