Eventually constant intertwining linear maps between complete locally convex spaces

Starting from Sinclair’s 1976 work [6] on automatic continuity of linear operators on Banach spaces, we prove that sequences of intertwining continuous linear maps are eventually constant with respect to the separating space of a fixed linear map. Our proof uses a gliding hump argument. We also consider aspects of continuity of linear functions between locally convex spaces and prove that such a linear function T from the locally convex space X to the locally convex space Y is continuous whenever the separating space G(T) is the zero vector in Y and for which X and Y satisfy conditions for a closed graph theorem.