In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn. Theorem The completeness of enables the following short proof, using the Baire category theorem. There are also simple proofs not using the Baire theorem . Corollaries The above corollary does claim that converges to in operator norm, that is, uniformly on bounded sets. However, since is bounded in operator norm, and the limit operator is continuous, a standard "" estimate shows that converges to uniformly on sets. Indeed, the elements of define a pointwise bounded family of continuous linear forms on the Banach space which is the continuous dual space of By the uniform boundedness principle, the norms of elements of as functionals on that is, norms in the second dual are bounded. But for every the norm in the second dual coincides with the norm in by a consequence of the Hahn–Banach theorem. Let denote the continuous operators from to endowed with the operator norm. If the collection is unbounded in then the uniform boundedness principle implies: In fact, is dense in The complement of in is the countable union of closed sets By the argument used in proving the theorem, each is nowhere dense, i.e. the subset is . Therefore is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called or ) are dense. Such reasoning leads to the , which can be formulated as follows: Example: pointwise convergence of Fourier series 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 is the -th Dirichlet kernel. Fix and consider the convergence of The functional defined