any balanced open neighborhood of the origin in such that The sets form an increasing (meaning implies ) cover of the compact space so there exists some such that (and thus ). It will be shown that for every thus demonstrating that is uniformly bounded in and completing the proof. So fix and Let The convexity of guarantees and moreover, since Thus which is a subset of Since is balanced and we have which combined with gives Finally, and imply as desired. Q.E.D. Since every singleton subset of is also a bounded subset, it follows that if is an equicontinuous set of continuous linear operators between two topological vector spaces and (not necessarily Hausdorff or locally convex), then the orbit of every is a bounded subset of Bounded subsets of topological modules The definition of bounded sets can be generalized to topological modules. A subset of a topological module over a topological ring is bounded if for any neighborhood of there exists a neighborhood of such that See also References Notes Bibliography Topological vector spaces