In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
Together with the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
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 operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear o ...
s (and thus
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
s) whose domain is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, pointwise boundedness is equivalent to uniform boundedness in
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
.
The theorem was first published in 1927 by
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original ...
and
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Unive ...
, but it was also proven independently by
Hans Hahn.
Theorem
The completeness of
enables the following short proof, using the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
.
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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
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
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
.
Let
denote the continuous operators from
to
endowed with the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
.
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
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
, 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
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, and let
be the Banach space of continuous functions on
with the
uniform norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when the ...
. 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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
is defined by
and the ''N''-th symmetric partial sum is
where
is the
-th
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any nonneg ...
. 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
(however, the Fourier series of a continuous function
converges to
for almost every
by
Carleson's theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the res ...
).
Generalizations
In a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS)
"bounded subset" refers specifically to the notion of a
von Neumann bounded subset. If
happens to also be a normed or
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
, say with
(semi)norm then a subset
is (von Neumann) bounded if and only if it is , which by definition means
Barrelled spaces
Attempts to find classes of
locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
s on which the uniform boundedness principle holds eventually led to
barrelled space
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 b ...
s.
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
Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of subsets of a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is said to be in
if there exists some
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
such that
which happens if and only if
is a bounded subset of
;
if
is a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
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
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
is used in the following version of the uniform bounded principle, the domain
is assumed to be a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
.
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 (meager) in
(and the same is thus also true of all its subsets).
Consequently, any vector subspace of a TVS
that is of the second category (nonmeager) in
must be a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of
(since otherwise its closure in
would a closed proper vector subspace of
and thus of the first category).
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
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
and the codomain is a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
then
Complete metrizable domain
proves a weaker form of this theorem with
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
s rather than the usual Banach spaces.
See also
*
*
Notes
Citations
Bibliography
* .
*
*
* .
*
*
*
* .
*
*
*
* .
* .
*
*
{{Boundedness and bornology
Articles containing proofs
Functional analysis
Mathematical principles
Theorems in functional analysis