Resumen de On the three-space property for locally convex spaces

Giorgio Metafune, V. B. Moscatelli

• The purpose of this paper is to prove that some familiar classes of locally convex spaces and, in particular, of Fréchet spaces, have the so-called three-space property, and also to give counterexamples showing that some important classes, which are often encountered in the applications, do not enjoy the above property.\newline We use standard notions and results from the theory of locally convex spaces, for which we refer to [4] and [3]. We also make reference to [7] for what concerns operator ideals on Banach spaces, ideals of locally convex spaces (space ideals) and Grothendieck space ideals. Here we confine ourselves to recalling that space ideals with the three-space property are termed \textit{three-space} ideals. This means that a class $\mathcal{C}$ of locally convex spaces is a three-space ideal if and only if the following conditions are satisfied: \begin{enumerate}[(i)] \item The finite-dimensioanl spaces belong to $\mathcal{C}$; \item $\mathcal{C}$ is stable under isomorphisms; \item If $E\epsilon\mathcal{C}$ and $F$ is a complemented subspace of $E$, then $F\epsilon\mathcal{C}$; \item Three-space property: if $F$ is a subspace of $E$ such that $F,E/F\epsilon\mathcal{C}$, then $E\epsilon\mathcal{C}$.\end{enumerate}