A theorem stating that pointwise boundedness implies uniform boundedness
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
Uniform Boundedness Principle — Let
be a Banach space,
a normed vector space and
the space of all continuous linear operators from
into
. Suppose that
is a collection of continuous linear operators from
to
If, for every
,
![{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed32dea213b322e6a6a595ec9a4c561b3caab7b)
then
![{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}=\sup _{\stackrel {T\in F,}{\|x\|\leq 1}}\|T(x)\|_{Y}<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b71f6d6fe12e14347d26d98d51d10cadf7a3b45c)
In the case that
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is not the trivial vector space, then the semi-inequality used in the supremum of the first term in this last chain of equalities (which has
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
range over the closed unit
ball) may be replaced by a proper equality (which has
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
range over the closed unit
sphere).
The completeness of
enables the following short proof, using the Baire category theorem.
Proof Let X be a Banach space. Suppose that for every
![{\displaystyle \sup _{T\in F}\|T(x)\|_{Y}<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c24783fcf959c11e8b9748062838da1b5df15f)
For every integer
let
![{\displaystyle X_{n}=\left\{x\in X\ :\ \sup _{T\in F}\|T(x)\|_{Y}\leq n\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8c2483751431539c47da626bc9f2afa65adcf1)
Each set
is a closed set and by the assumption,
![{\displaystyle \bigcup _{n\in \mathbb {N} }X_{n}=X\neq \varnothing .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06ace97a13f52679adc8cb5c41a7f3730597551c)
By the Baire category theorem for the non-empty complete metric space
there exists some
such that
has non-empty interior; that is, there exist
and
such that
![{\displaystyle {\overline {B_{\varepsilon }(x_{0})}}~:=~\left\{x\in X\,:\,\|x-x_{0}\|\leq \varepsilon \right\}~\subseteq ~X_{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4bb0950788481fc4012987697c845d733b4d62)
Let
with
and
Then:
![{\displaystyle {\begin{aligned}\|T(u)\|_{Y}&=\varepsilon ^{-1}\left\|T\left(x_{0}+\varepsilon u\right)-T\left(x_{0}\right)\right\|_{Y}&[{\text{by linearity of }}T]\\&\leq \varepsilon ^{-1}\left(\left\|T(x_{0}+\varepsilon u)\right\|_{Y}+\left\|T(x_{0})\right\|_{Y}\right)\\&\leq \varepsilon ^{-1}(m+m).&[{\text{since }}\ x_{0}+\varepsilon u,\ x_{0}\in X_{m}]\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/586805bd601cab526a89e01c79fd8eb5b805a3ed)
Taking the supremum over
in the unit ball of
and over
it follows that
![{\displaystyle \sup _{T\in F}\|T\|_{B(X,Y)}~\leq ~2\varepsilon ^{-1}m~<~\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eca644f1f8ca81a01a08a7674dcceac4aff9679)
There are also simple proofs not using the Baire theorem (Sokal 2011).
Corollaries
The above corollary does not 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 compact sets.
Proof 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.
By uniform boundedness principle, let
be a uniform upper bound on the operator norms.
Fix any compact
. Then for any
, finitely cover (use compactness)
by a finite set of open balls
of radius
Since
pointwise on each of
, for all large
,
for all
.
Then by triangle inequality, we find for all large
,
.
Corollary — Any weakly bounded subset
in a normed space
is bounded.
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:
![{\displaystyle R=\left\{x\in X\ :\ \sup \nolimits _{T\in F}\|Tx\|_{Y}=\infty \right\}\neq \varnothing .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82fed5720d5d1442e57a9bf8e55c588ab9f0ae57)
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 of first category. Therefore
is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called comeagre or residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows:
Proof The complement of
is the countable union
![{\displaystyle \bigcup _{n,m}\left\{x\in X\ :\ \sup _{T\in F_{n}}\|Tx\|_{Y_{n}}\leq m\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c10cbdf68f8436a257ff52621816fbf68780a829)
of sets of first category. Therefore, its residual set
![{\displaystyle R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
is dense.
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
![{\displaystyle \sum _{k\in \mathbb {Z} }{\hat {f}}(k)e^{ikx}=\sum _{k\in \mathbb {Z} }{\frac {1}{2\pi }}\left(\int _{0}^{2\pi }f(t)e^{-ikt}dt\right)e^{ikx},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2c1b0d64046109428b35f9c6baeb7f8297f17f)
and the
N-th symmetric partial sum is
![{\displaystyle S_{N}(f)(x)=\sum _{k=-N}^{N}{\hat {f}}(k)e^{ikx}={\frac {1}{2\pi }}\int _{0}^{2\pi }f(t)D_{N}(x-t)\,dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6422f2ecda7b2a9af0ee02b01a1adccd70dd175)
where
![{\displaystyle D_{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69fcf2425a1e9f9ef9979b837f32bdfbcd4beff5)
is the
![{\displaystyle N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
-th
Dirichlet kernel. Fix
![{\displaystyle x\in \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d049929c2e82026ec84d3c2300c3fc905dce4633)
and consider the convergence of
![{\displaystyle \left\{S_{N}(f)(x)\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0788b7652957a7a47255951503c4dda8c169b6)
The functional
![{\displaystyle \varphi _{N,x}:C(\mathbb {T} )\to \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce2b145a725c9141bcd3755dba5dd470a739bf3)
defined by
![{\displaystyle \varphi _{N,x}(f)=S_{N}(f)(x),\qquad f\in C(\mathbb {T} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64008973d0a74b6d0d454a7bf558dffcccddfd13)
is bounded. The norm of
![{\displaystyle \varphi _{N,x},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33deafcbe90449cf0a53aee9370ece9036e07551)
in the dual of
![{\displaystyle C(\mathbb {T} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/254d0a6a57f1f33c8cac4d20b5d2ddeabd8294f9)
is the norm of the signed measure
![{\displaystyle (2(2\pi )^{-1}D_{N}(x-t)dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8c7390c6429a78c7568d3ec1b0bfc8d0cc0899)
namely
![{\displaystyle \left\|\varphi _{N,x}\right\|={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{L^{1}(\mathbb {T} )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3c162e6583ded595011d7026c05c14a1dfe315)
It can be verified that
![{\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|D_{N}(t)|\,dt\geq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left|\sin \left((N+{\tfrac {1}{2}})t\right)\right|}{t/2}}\,dt\to \infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e008634f407325c420c1b9838e3e770a980603a0)
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).
Generalizations
In a topological vector space (TVS)
"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 (von Neumann) bounded if and only if it is norm bounded, which by definition means
Barrelled 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 (Bourbaki 1987, Theorem III.2.1):
Uniform boundedness in topological vector spaces
A family
of subsets of a topological vector space
is said to be uniformly bounded in
if there exists some bounded subset
of
such that
![{\displaystyle B\subseteq D\quad {\text{ for every }}B\in {\mathcal {B}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bee21ec6c0c8da5bae2ec6ba6a8413e5c5af3f3)
which happens if and only if
![{\displaystyle \bigcup _{B\in {\mathcal {B}}}B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9455ea3c6fa3783128e9a0bb751df65d83e8d3a)
is a bounded subset of
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
; if
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is a
normed space then this happens if and only if there exists some real
![{\displaystyle M\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/069c80bca00b08466c7ef8ce37a46a56035dcef8)
such that
![{\textstyle \sup _{\stackrel {b\in B}{B\in {\mathcal {B}}}}\|b\|\leq M.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1279c9f873b7164619ab42a4bdf95142b6c5c23)
In particular, if
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is a family of maps from
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
to
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
and if
![{\displaystyle C\subseteq X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a98beab5a3637ffa2321ef617c02a3b5b26ffd3a)
then the family
![{\displaystyle \{h(C):h\in H\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d78886915ad4a3b36d630a7b3524453463201a0a)
is uniformly bounded in
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
if and only if there exists some bounded subset
![{\displaystyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6)
of
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
such that
![{\displaystyle h(C)\subseteq D{\text{ for all }}h\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65f6627c29f6530abacdbbc92259a634763120ec)
which happens if and only if
![{\textstyle H(C):=\bigcup _{h\in H}h(C)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/723108ba56f8026a3fe2241f8b035467f4fcf6e0)
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 not assumed to be a Baire space.
Theorem — Let
be a set of continuous linear operators between two topological vector spaces
and
(not necessarily Hausdorff or locally convex). For every
denote the orbit of
by
![{\displaystyle H(x):=\{h(x):h\in H\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5a48fe43e6006f982590bd2c3433846c2ef379)
and let
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
denote the set of all
![{\displaystyle x\in X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
whose orbit
![{\displaystyle H(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f241aa7195bebab9d0a3c248ea97ef0c78b1ac)
is a bounded subset of
![{\displaystyle Y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61)
If
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
is of the
second category (that is, nonmeager) in
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
then
![{\displaystyle B=X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/764e9d661309f4d92ca2e43d1aa6573f92a12dd4)
and
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is equicontinuous.
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 of
(since otherwise its closure in
would a closed proper vector subspace of
and thus of the first category).
Proof Proof that
is equicontinuous:
Let
be balanced neighborhoods of the origin in
satisfying
It must be shown that there exists a neighborhood
of the origin in
such that
for every
Let
![{\displaystyle C~:=~\bigcap _{h\in H}h^{-1}\left({\overline {V}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc6edd3e14d2b320dbc6be25f20e6b792b31895)
which is a closed subset of
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
(because it is an intersection of closed subsets) that for every
![{\displaystyle h\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad91aec7b3a20bc5c6ca74a374ef9032b1d8be)
also satisfies
![{\displaystyle h(C)\subseteq {\overline {V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d84baf674bd46b7282d4d8ac20905f0e100433)
and
![{\displaystyle h(C-C)~=~h(C)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~=~{\overline {V}}+{\overline {V}}~\subseteq ~W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d81f9abccff399925c0bcd55621d54bb1a70410)
(as will be shown, the set
![{\displaystyle C-C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a81c013f8303e973aab22cb215624529f3b4c26d)
is in fact a neighborhood of the origin in
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
because the topological interior of
![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
in
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is not empty). If
![{\displaystyle b\in B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86)
then
![{\displaystyle H(b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/810ab14fd6bf72cf37e744fe2f1dce945b78d7b6)
being bounded in
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
implies that there exists some integer
![{\displaystyle n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
such that
![{\displaystyle H(b)\subseteq nV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/743882e61d55ed56344831a2162982da767c5939)
so if
![{\displaystyle h\in H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad91aec7b3a20bc5c6ca74a374ef9032b1d8be)
then
![{\displaystyle b~\in ~h^{-1}\left(nV\right)~=~nh^{-1}(V).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae29ca75ed0c1bd8972069e38e6561c9ac9b556e)
Since
![{\displaystyle h\in H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675a79e26028d91d97f4e2ce279c314b0f194c1e)
was arbitrary,
![{\displaystyle b~\in ~\bigcap _{h\in H}nh^{-1}(V)~=~n\bigcap _{h\in H}h^{-1}(V)~\subseteq ~nC.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9abed8c696a34e9223f51b8bb3d3a05bd23af6d)
This proves that
![{\displaystyle B~\subseteq ~\bigcup _{n\in \mathbb {N} }nC.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/126477adaf4cfecd69f494ef0503453304ed28a8)
Because
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
is of the second category in
![{\displaystyle X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df)
the same must be true of at least one of the sets
![{\displaystyle nC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b586b1549993466229bb5bb0d266a04e0c0f8bb2)
for some
![{\displaystyle n\in \mathbb {N} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b82f4024a32a795b10f00971e03ad88ebaa367)
The map
![{\displaystyle X\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219)
defined by
![{\textstyle x\mapsto {\frac {1}{n}}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05b7ba3a6e3dc377d2fc3d2f1f1566c4e56422ca)
is a (
surjective)
homeomorphism, so the set
![{\textstyle {\frac {1}{n}}(nC)=C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9c6aba5aeaef7bc70a745fb3519391a7149641)
is necessarily of the second category in
![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
Because
![{\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
is closed and of the second category in
![{\displaystyle X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df)
its
topological interior in
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is not empty. Pick
![{\displaystyle c\in \operatorname {Int} _{X}C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c4991d1b9cb6bb978dee75316807d8d29832cf)
Because the map
![{\displaystyle X\to X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83d8a6029587ee9b365bdeab1e2f4b7c469b0219)
defined by
![{\displaystyle x\mapsto c-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eae78c484db504aa86eb8bc887331205bc2fa385)
is a homeomorphism, the set
![{\displaystyle N~:=~c-\operatorname {Int} _{X}C~=~\operatorname {Int} _{X}(c-C)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d76f09378b80607de5933debbe0bfb9e9b8d76a)
is a neighborhood of
![{\displaystyle 0=c-c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1beff98c15a5089580eb4fb402960ac643ef7e30)
in
![{\displaystyle X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba32eeb405f7f5f2bac1eb12987c47d2fd42df)
which implies that the same is true of its superset
![{\displaystyle C-C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79bb73bba5cccf0216f292d60208f418deb06318)
And so for every
![{\displaystyle h(N)~\subseteq ~h(c-C)~=~h(c)-h(C)~\subseteq ~{\overline {V}}-{\overline {V}}~\subseteq ~W.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62d65713ffde211e1d50d8030e44de4480fde329)
This proves that
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is equicontinuous. Q.E.D.
Proof that
:
Because
is equicontinuous, if
is bounded in
then
is uniformly bounded in
In particular, for any
because
is a bounded subset of
is a uniformly bounded subset of
Thus
Q.E.D.
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.
Theorem — Suppose that
is a sequence of continuous linear maps between two topological vector spaces
and
- If the set
of all
for which
is a Cauchy sequence in
is of the second category in
then ![{\displaystyle C=X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22eacb4374a77c1272969a123b43e2adce7c61e4)
- If the set
of all
at which the limit
exists in
is of the second category in
and if
is a complete metrizable topological vector space (such as a Fréchet space or an F-space), then
and
is a continuous linear map.
Theorem — If
is a sequence of continuous linear maps from an F-space
into a Hausdorff topological vector space
such that for every
the limit
![{\displaystyle h(x)~:=~\lim _{n\to \infty }h_{n}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44aacc6abe2e5149808f2389ca1e4bf615542969)
exists in
![{\displaystyle Y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3765557b7effa1a5f2f4dce9c80a25973b7009f)
then
![{\displaystyle h:X\to Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d91a802daa0df93b0c6fcc9e2065fac31ce08e3b)
is a continuous linear map and the maps
![{\displaystyle h,h_{1},h_{2},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9af2d6e7240d9fff7bd8afedf5e26a93afe9b3e9)
are equicontinuous.
If in addition the domain is a Banach space and the codomain is a normed space then
Complete metrizable domain
Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces.
Theorem — Let
be a set of continuous linear operators from a complete metrizable topological vector space
(such as a Fréchet space or an F-space) into a Hausdorff topological vector space
If for every
the orbit
![{\displaystyle H(x):=\{h(x):h\in H\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5a48fe43e6006f982590bd2c3433846c2ef379)
is a bounded subset of
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
then
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is equicontinuous.
So in particular, if
is also a normed space and if
![{\displaystyle \sup _{h\in H}\|h(x)\|<\infty \quad {\text{ for every }}x\in X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/063d7efd8f73de903185cdaa718e39a89d37e8b6)
then
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is equicontinuous.
See also
Notes
Citations
Bibliography
- Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF), Fundamenta Mathematicae, 9: 50–61, doi:10.4064/fm-9-1-50-61. (in French)
- Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.
- Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
- Dieudonné, Jean (1970), Treatise on analysis, Volume 2, Academic Press.
- Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. Vol. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665.
- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Rudin, Walter (1966), Real and complex analysis, McGraw-Hill.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
- Shtern, A.I. (2001) [1994], "Uniform boundedness principle", Encyclopedia of Mathematics, EMS Press.
- Sokal, Alan (2011), "A really simple elementary proof of the uniform boundedness theorem", Amer. Math. Monthly, 118 (5): 450–452, arXiv:1005.1585, doi:10.4169/amer.math.monthly.118.05.450, S2CID 41853641.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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