Topological Modules/Barrelled spaces

Proposition (pointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear):

Let be a barrelled LCTVS over a field , let be a locally closed Hausdorff TVS over the same field, and suppose that a net () of linear and continuous functions is given. Suppose further that is a function such that


Then is itself a linear and continuous functional.

Proof: First note that is linear, since whenever and , we have

since is a Hausdorff space, where limits are well-defined, and by continuity of addition. Then note that is continuous, since for all the set is bounded, so that the Banach—Steinhaus theorem applies and the family is uniformly bounded. Hence, suppose that is a closed neighbourhood of the origin. By uniform boundedness, select to be an open neighbourhood of the origin so that


We conclude that , since closed sets contain their net limits. We conclude since is locally closed, so that represents a generic neighbourhood.