Theorem. Let X be a Banach space and Y a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X, then The completeness of X enables the following short proof, using the Baire category theorem. There are also simple proofs not using the Baire theorem.
Corollaries
Note it is not claimed above that Tn converges to T in operator norm, i.e. uniformly on bounded sets. Indeed, the elements of S define a pointwise bounded family of continuous linear forms on the Banach space X = Y*, continuous dual of Y. By the uniform boundedness principle, the norms of elements of S, as functionals on X, that is, norms in the second dual Y**, are bounded. But for every s in S, the norm in the second dual coincides with the norm in Y, by a consequence of the Hahn–Banach theorem. Let L denote the continuous operators from X to Y, with the operator norm. If the collection F is unbounded in L, then by the uniform boundedness principle, we have: In fact, R is dense in X. The complement of R in X is the countable union of closed sets ∪Xn. By the argument used in proving the theorem, each Xn is nowhere dense, i.e. the subset ∪Xn is of first category. Therefore R is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows: Theorem. Let X be a Banach space, a sequence of normed vector spaces, and Fn an unbounded family in L. Then the set is a residual set, and thus dense in X.
Let be the circle, and let be the Banach space of continuous functions on, with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in for which the Fourier series does not converge pointwise. For, its Fourier series is defined by and the N-th symmetric partial sum is where DN is the N-th Dirichlet kernel. Fix and consider the convergence of. The functional defined by is bounded. The norm of φN,x, in the dual of, is the norm of the signed measure−1DN dt, namely One can verify 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 x is dense in. More can be concluded by applying the principle of condensation of singularities. Let be a dense sequence in. Define φN,xm 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 xm is dense in .
Generalisations
The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalised version of the theorem holds : Alternatively, the statement also holds whenever X is a Baire space and Y is a locally convex space. proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically, then the family H is equicontinuous.
