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.
The completeness of enables the following short proof, using the .
There are also simple proofs not using the Baire theorem .
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:
Let be the circle, and let be the Banach space of continuous functions on with the uniform norm.
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In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them. Barrelled spaces were introduced by .
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
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Concepts de base de l'analyse fonctionnelle linéaire: opérateurs bornés, opérateurs compacts, théorie spectrale pour les opérateurs symétriques et compacts, le théorème de Hahn-Banach, les théorèmes d
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