Continuity of homomorphisms of Banach \(G\)-modules (Q1068957)

Let G be a locally compact abelian group. Let X and Y be two Banach G- modules, and let S be a discontinuous G-module homomorphism from X into Y. Then there exists a submodule of Y on which G acts by scalar multiplication, say by the character \(\chi\), and there exists a discontinuous \(\chi\)-covariant linear form on X. Put in another way, unless there is such a submodule of Y and such a form on X, every G- module homomorphism is continuous. The proof of this theorem in the present paper depends on showing that the ''separating space'' of S is a finite sum of scalar submodules of Y. [The ''separating space'' is defined in \textit{A. M. Sinclair}, Automatic continuity of linear operators (1976; Zbl 0313.47029).]