In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis.
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.
enables the following short proof, using the Baire category theorem.
By the Baire category theorem for the non-empty complete metric space
There are also simple proofs not using the Baire theorem (Sokal 2011).
then these pointwise limits define a bounded linear operator
in operator norm, that is, uniformly on bounded sets.
Essentially the same as that of the proof that a pointwise convergent sequence of equicontinuous functions on a compact set converges to a continuous function.
be a uniform upper bound on the operator norms.
define a pointwise bounded family of continuous linear forms on the Banach space
By the uniform boundedness principle, the norms of elements of
is the complement of a subset of first category in a Baire space.
Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows: Theorem — Let
Using the uniform boundedness principle, one can show that there exists an element in
the set of continuous functions whose Fourier series diverges at
More can be concluded by applying the principle of condensation of singularities.
The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each
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 (Bourbaki 1987, Theorem III.2.1): Theorem — Given a barrelled space
then any family of pointwise bounded continuous linear mappings from
be a set of continuous linear operators between two topological vector spaces
if any of the following conditions are satisfied: Although the notion of a nonmeager set is used in the following version of the uniform bounded principle, the domain
be a set of continuous linear operators between two topological vector spaces
[3] So in particular, every proper vector subspace that is closed is nowhere dense in
The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous.
is a sequence of continuous linear maps between two topological vector spaces
is a sequence of continuous linear maps from an F-space
Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces.
be a set of continuous linear operators from a complete metrizable topological vector space