Uniform boundedness principle
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 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 first inequality states that the functionals in are pointwise bounded while the second states that they are uniformly bounded.The second supremum always equals
and if is not the trivial vector space then closed unit ball can be replaced with the unit sphere
The completeness of the Banach space 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 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 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 .
Generalizations
In a topological vector space "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 bounded if and only if it is, which by definition meansBarrelled 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 thatwhich 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 in .
Consequently, any vector subspace of a TVS that is of the second category in must be a dense subset of .
Sequences of continuous linear maps
The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous.If in addition the domain is a Banach space and the codomain is a normed space then