by is bounded. The norm of in the dual of is the norm of the signed measure namely It can be verified that So the collection is unbounded in the dual of Therefore, by the uniform boundedness principle, for any the set of continuous functions whose Fourier series diverges at is dense in More can be concluded by applying the principle of condensation of singularities. Let be a dense sequence in Define in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each is dense in (however, the Fourier series of a continuous function converges to for almost every by Carleson's theorem). Generalizations In a topological vector space (TVS) "bounded subset" refers specifically to the notion of a von Neumann bounded subset. If happens to also be a normed or seminormed space, say with (semi)norm then a subset is (von Neumann) bounded if and only if it is , which by definition means Barrelled spaces Attempts to find classes of locally convex topological vector spaces on which the uniform boundedness principle holds eventually led to barrelled spaces. That is, the least restrictive setting for the uniform boundedness principle is a barrelled space, where the following generalized version of the theorem holds : Uniform boundedness in topological vector spaces A family of subsets of a topological vector space is said to be in if there exists some bounded subset of such that which happens if and only if is a bounded subset of ; if is a normed space then this happens if and only if there exists some real such that In particular, if is a family of maps from to and if then the family is uniformly bounded in if and only if there exists some bounded subset of such that which happens if and only if is a bounded subset of Generalizations involving nonmeager subsets Although the notion of a nonmeager set is used in the following version of the uniform bounded principle, the domain is assumed to be a Baire space. Every proper vector subspace of a TVS has an empty interior in So in particular, every proper vector subspace that is closed is nowhere dense in and thus of the first category (meager) in (and the same is thus also true of all its subsets). Consequently, any vector subspace of a