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{{Short description|A theorem stating that pointwise boundedness implies uniform boundedness}}
{{for multi|the definition of uniformly bounded functions|Uniform boundedness|the conjectures in number theory and algebraic geometry|Uniform boundedness conjecture (disambiguation)}}
In [[Mathematics|mathematics]], the '''uniform boundedness principle''' or '''Banach–Steinhaus theorem''' is one of the fundamental results in [[Functional analysis|functional analysis]]. 
Together with the [[Hahn–Banach theorem]] and the [[Open mapping theorem (functional analysis)|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|continuous linear operator]]s (and thus [[Bounded operator|bounded operator]]s) whose domain is a [[Banach space]], pointwise [[Bounded set|boundedness]] is equivalent to uniform boundedness in [[Operator norm|operator norm]].

The theorem was first published in 1927 by [[Biography:Stefan Banach|Stefan Banach]] and [[Biography:Hugo Steinhaus|Hugo Steinhaus]], but it was also proven independently by [[Biography:Hans Hahn (mathematician)|Hans Hahn]].

==Theorem==

{{math theorem|name=Uniform Boundedness Principle|math_statement=
Let <math>X</math> be a [[Banach space]], <math>Y</math> a [[Normed vector space|normed vector space]] and <math>B(X,Y)</math> the space of all continuous linear operators from <math>X</math> into <math>Y</math>. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y.</math> If, for every <math>x \in X</math>,
<math display=block>\sup_{T \in F} \|T(x)\|_Y  < \infty,</math>
then
<math display=block>\sup_{T \in F} \|T\|_{B(X,Y)} = \sup_{\stackrel{T \in F,}{\|x\|\leq 1}} \|T(x)\|_Y  < \infty.</math>
In the case that <math>X</math> 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 <math>x</math> range over the closed unit ''ball'') may be replaced by a proper equality (which has <math>x</math> range over the closed unit ''sphere'').
}}

The completeness of <math>X</math> enables the following short proof, using the [[Baire category theorem]].

{{math proof | proof =
Let {{mvar|X}} be a Banach space. 
Suppose that for every <math>x \in X,</math>
<math display=block>\sup_{T \in F} \|T(x)\|_Y < \infty.</math>

For every integer <math>n \in \N,</math> let
<math display=block>X_n = \left\{x \in X \ : \ \sup_{T \in F} \|T (x)\|_Y \leq n \right\}.</math>

Each set <math>X_n</math> is a [[Closed set|closed set]] and by the assumption,
<math display=block>\bigcup_{n \in \N} X_n = X \neq \varnothing.</math>

By the [[Baire category theorem]] for the non-empty [[Complete metric space|complete metric space]] <math>X,</math> there exists some <math>m \in \N</math> such that 
<math>X_m</math> has non-empty [[Interior (topology)|interior]]; that is, there exist <math>x_0 \in X_m</math> and <math>\varepsilon > 0</math> such that
<math display=block>\overline{B_\varepsilon (x_0)} ~:=~ \left\{x \in X \,:\, \|x - x_0\| \leq \varepsilon \right\} ~\subseteq~ X_m.</math>

Let <math>u \in X</math> with <math>\|u\| \leq 1</math> and <math>T \in F.</math>
Then:
<math display=block>\begin{align}
\|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{align}</math>

Taking the supremum over <math>u</math> in the unit ball of <math>X</math> and over <math>T \in F</math> it follows that
<math display=block>\sup_{T \in F} \|T\|_{B(X,Y)} ~\leq~ 2 \varepsilon^{-1} m ~ < ~ \infty.</math>
}}

There are also simple proofs not using the Baire theorem {{harv|Sokal|2011}}.

==Corollaries==

{{math theorem|name=Corollary | math_statement=
If a sequence of bounded operators <math>\left(T_n\right)</math> converges pointwise, that is, the limit of <math>\left(T_n(x)\right)</math> exists for all <math>x \in X,</math> then these pointwise limits define a bounded linear operator <math>T.</math>
}}

The above corollary does {{em|not}} claim that <math>T_n</math> converges to <math>T</math> in operator norm, that is, uniformly on bounded sets.  However, since <math>\left\{T_n\right\}</math> is bounded in operator norm, and the limit operator <math>T</math> is continuous, a standard "<math>3\varepsilon</math>" estimate shows that <math>T_n</math> converges to <math>T</math> uniformly on {{em|compact}} sets. 

{{math proof | proof =
Essentially the same as that of the proof that a pointwise convergent sequence of uniformly continuous functions on a compact set converges to a continuous function.

By uniform boundedness principle, let <math>M = \max\{\sup_n \|T_n\|, T\}</math> be a uniform upper bound on the operator norms. 

Fix any compact <math>K\subset X</math>. Then for any <math>\epsilon > 0</math>, finitely cover (use compactness) <math>K</math> by a finite set of open balls <math>\{B(x_i, r)\}_{i=1, ..., N}</math> of radius <math>r = \frac{\epsilon}{M}</math>

Since <math>T_n \to T</math> pointwise on each of <math>x_1, ..., x_N</math>, for all large <math>n</math>, <math>\|T_n(x_i) - T(x_i)\|\leq \epsilon</math> for all <math>i= 1,..., N</math>.

Then by triangle inequality, we find for all large <math>n</math>, <math>\forall x\in K, \|T_n(x) - T(x)\|\leq 3\epsilon</math>.
}}

{{math theorem | name=Corollary | math_statement=
Any weakly bounded subset <math>S \subseteq Y</math> in a normed space <math>Y</math> is bounded.
}}

Indeed, the elements of <math>S</math> define a pointwise bounded family of continuous linear forms on the Banach space <math>X := Y',</math> which is the continuous dual space of <math>Y.</math> 
By the uniform boundedness principle, the norms of elements of <math>S,</math> as functionals on <math>X,</math> that is, norms in the second dual <math>Y'',</math> are bounded. 
But for every <math>s \in S,</math> the norm in the second dual coincides with the norm in <math>Y,</math> by a consequence of the [[Hahn–Banach theorem]].

Let <math>L(X, Y)</math> denote the continuous operators from <math>X</math> to <math>Y,</math> endowed with the [[Operator norm|operator norm]]. 
If the collection <math>F</math> is unbounded in <math>L(X, Y),</math> then the uniform boundedness principle implies:
<math display=block>R = \left \{x \in X  \ : \ \sup\nolimits_{T \in F} \|Tx\|_Y = \infty \right\} \neq \varnothing.</math>

In fact, <math>R</math> is dense in <math>X.</math> The complement of <math>R</math> in <math>X</math> is the countable union of closed sets <math display="inline">\bigcup X_n.</math>
By the argument used in proving the theorem, each <math>X_n</math> is nowhere dense, i.e. the subset <math display="inline">\bigcup X_n</math> is {{em|of first category}}. 
Therefore <math>R</math> is the complement of a subset of first category in a  Baire space. By definition of a Baire space, such sets (called {{em|comeagre}} or {{em|residual sets}}) are dense. 
Such reasoning leads to the {{em|principle of condensation of singularities}}, which can be formulated as follows:

{{math theorem | math_statement=
Let <math>X</math> be a Banach space, <math>\left(Y_n\right)</math> a sequence of normed vector spaces, and for every <math>n,</math> let <math>F_n</math> an unbounded family in <math>L\left(X, Y_n\right).</math> Then the set
<math display=block>R := \left\{x \in X \ : \ \text{ for all } n \in \N , \sup_{T \in F_n} \|Tx\|_{Y_n} = \infty\right\}</math>
is a residual set, and thus dense in <math>X.</math>
}}

{{math proof| proof = 
The complement of <math>R</math> is the countable union
<math display=block>\bigcup_{n,m} \left\{x \in X \ : \ \sup_{T \in F_n} \|Tx\|_{Y_n} \leq m\right\}</math>
of sets of first category. Therefore, its residual set <math>R</math> is dense.
}}

==Example: pointwise convergence of Fourier series==

Let <math>\mathbb{T}</math> be the [[Circle group|circle]], and let <math>C(\mathbb{T})</math> be the Banach space of continuous functions on <math>\mathbb{T},</math> with the [[Uniform norm|uniform norm]]. Using the uniform boundedness principle, one can show that there exists an element in <math>C(\mathbb{T})</math> for which the Fourier series does not converge pointwise.

For <math>f \in C(\mathbb{T}),</math> its [[Fourier series]] is defined by
<math display=block>\sum_{k \in \Z} \hat{f}(k) e^{ikx} = \sum_{k \in \Z} \frac{1}{2\pi} \left (\int_0 ^{2 \pi} f(t) e^{-ikt} dt \right) e^{ikx},</math>
and the ''N''-th symmetric partial sum is
<math display=block>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,</math>
where <math>D_N</math> is the <math>N</math>-th [[Dirichlet kernel]]. Fix <math>x \in \mathbb{T}</math> and consider the convergence of <math>\left\{S_N(f)(x)\right\}.</math> 
The functional <math>\varphi_{N,x} : C(\mathbb{T}) \to \Complex</math> defined by
<math display=block>\varphi_{N, x}(f) =  S_N(f)(x), \qquad f \in C(\mathbb{T}),</math>
is bounded. 
The norm of <math>\varphi_{N,x},</math> in the dual of <math>C(\mathbb{T}),</math> is the norm of the signed measure <math>(2(2 \pi)^{-1} D_N(x - t) d t,</math> namely
<math display=block>\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})}.</math>

It can be verified that
<math display=block>\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.</math>

So the collection <math>\left(\varphi_{N, x}\right)</math> is unbounded in <math>C(\mathbb{T})^{\ast},</math> the dual of <math>C(\mathbb{T}).</math> 
Therefore, by the uniform boundedness principle, for any <math>x \in \mathbb{T},</math> the set of continuous functions whose Fourier series diverges at <math>x</math> is dense in <math>C(\mathbb{T}).</math>

More can be concluded by applying the principle of condensation of singularities. 
Let <math>\left(x_m\right)</math> be a dense sequence in <math>\mathbb{T}.</math> 
Define <math>\varphi_{N, x_m}</math> 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 <math>x_m</math>  is dense in <math>C(\mathbb{T})</math> (however, the Fourier series of a continuous function <math>f</math> converges to <math>f(x)</math> for almost every <math>x \in \mathbb{T},</math> by [[Carleson's theorem]]).

==Generalizations==

In a [[Topological vector space|topological vector space]] (TVS) <math>X,</math> "bounded subset" refers specifically to the notion of a [[Bounded set (topological vector space)|von Neumann bounded subset]]. If <math>X</math> happens to also be a normed or seminormed space, say with [[Seminorm|(semi)norm]] <math>\|\cdot\|,</math> then a subset <math>B</math> is (von Neumann) bounded if and only if it is {{em|norm bounded}}, which by definition means <math display="inline">\sup_{b \in B} \|b\| < \infty.</math>

===Barrelled spaces===
{{Main|Barrelled space}}

Attempts to find classes of [[Locally convex topological vector space|locally convex topological vector space]]s on which the uniform boundedness principle holds eventually led to [[Barrelled space|barrelled space]]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 {{harv|Bourbaki|1987|loc=Theorem III.2.1}}:

{{math theorem |math_statement=
Given a barrelled space <math>X</math> and a [[Locally convex topological vector space|locally convex space]] <math>Y,</math> then any family of pointwise bounded continuous linear mappings from <math>X</math> to <math>Y</math> is equicontinuous (and even uniformly equicontinuous).

Alternatively, the statement also holds whenever <math>X</math> is a [[Baire space]] and <math>Y</math> is a locally convex space.{{sfn|Shtern|2001}}
}}

===Uniform boundedness in topological vector spaces===


A [[Family of sets|family]] <math>\mathcal{B}</math> of subsets of a [[Topological vector space|topological vector space]] <math>Y</math> is said to be {{em|uniformly bounded}} in <math>Y,</math> if there exists some [[Bounded set (topological vector space)|bounded subset]] <math>D</math> of <math>Y</math> such that 
<math display=block>B \subseteq D \quad \text{ for every } B \in \mathcal{B},</math>
which happens if and only if 
<math display=block>\bigcup_{B \in \mathcal{B}} B</math> 
is a bounded subset of <math>Y</math>; 
if <math>Y</math> is a normed space then this happens if and only if there exists some real <math>M \geq 0</math> such that <math display="inline">\sup_{\stackrel{b \in B}{B \in \mathcal{B}}} \|b\| \leq M.</math> 
In particular, if <math>H</math> is a family of maps from <math>X</math> to <math>Y</math> and if <math>C \subseteq X</math> then the family <math>\{h(C) : h \in H\}</math> is uniformly bounded in <math>Y</math> if and only if there exists some bounded subset <math>D</math> of <math>Y</math> such that <math>h(C) \subseteq D \text{ for all } h \in H,</math> which happens if and only if <math display=inline>H(C) := \bigcup_{h \in H} h(C)</math> is a bounded subset of <math>Y.</math> 

{{math theorem | name = Proposition{{sfn|Rudin|1991|pp=42−47}} | math_statement=
Let <math>H \subseteq L(X, Y)</math> be a set of continuous linear operators between two [[Topological vector space|topological vector space]]s <math>X</math> and <math>Y</math> and let <math>C \subseteq X</math> be any [[Bounded set (topological vector space)|bounded subset]] of <math>X.</math> 
Then the [[Family of sets|family of sets]] <math>\{h(C) : h \in H\}</math> is uniformly bounded in <math>Y</math> if any of the following conditions are satisfied:
# <math>H</math> is equicontinuous.
# <math>C</math> is a [[Convex set|convex]] [[Compact space|compact]] Hausdorff subspace of <math>X</math> and for every <math>c \in C,</math> the orbit <math>H(c) := \{h(c) : h \in H\}</math> is a bounded subset of <math>Y.</math>
}}

===Generalizations involving nonmeager subsets===

Although the notion of a nonmeager set is used in the following version of the uniform bounded principle, the domain <math>X</math> is {{em|not}} assumed to be a [[Baire space]].

{{math theorem | name = Theorem{{sfn|Rudin|1991|pp=42−47}} | math_statement=
Let <math>H \subseteq L(X, Y)</math> be a set of [[Continuous linear operator|continuous linear operator]]s between two  [[Topological vector space|topological vector space]]s <math>X</math> and <math>Y</math> (not necessarily [[Hausdorff space|Hausdorff]] or locally convex). 
For every <math>x \in X,</math> denote the orbit of <math>x</math> by 
<math display=block>H(x) := \{h(x) : h \in H\}</math> 
and let <math>B</math> denote the set of all <math>x \in X</math> whose orbit <math>H(x)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y.</math> 
If <math>B</math> is of the second category (that is, nonmeager) in <math>X</math> then <math>B = X</math> and <math>H</math> is equicontinuous.
}}

Every proper vector subspace of a TVS <math>X</math> has an empty interior in <math>X.</math>{{sfn|Rudin|1991|p=46}} So in particular, every proper vector subspace that is closed is nowhere dense in <math>X</math> and thus of the first category (meager) in <math>X</math> (and the same is thus also true of all its subsets). 
Consequently, any vector subspace of a TVS <math>X</math> that is of the second category (nonmeager) in <math>X</math> must be a [[Dense set|dense subset]] of <math>X</math> (since otherwise its closure in <math>X</math> would a closed proper vector subspace of <math>X</math> and thus of the first category).{{sfn|Rudin|1991|p=46}} 

{{math proof | title = Proof{{sfn|Rudin|1991|pp=42−47}} | proof =
{{em|Proof that <math>H</math> is equicontinuous:}}

Let <math>W, V \subseteq Y</math> be [[Balanced set|balanced]] neighborhoods of the origin in <math>Y</math> satisfying <math>\overline{V} + \overline{V} \subseteq W.</math> It must be shown that there exists a neighborhood <math>N \subseteq X</math> of the origin in <math>X</math> such that <math>h(N) \subseteq W</math> for every <math>h \in H.</math> 
Let 
<math display=block>C ~:=~ \bigcap_{h \in H} h^{-1}\left(\overline{V}\right),</math>
which is a closed subset of <math>X</math> (because it is an intersection of closed subsets) that for every <math>h \in H,</math> also satisfies <math>h(C) \subseteq \overline{V}</math> and 
<math display=block>h(C - C) ~=~ h(C) - h(C) ~\subseteq~ \overline{V} - \overline{V} ~=~ \overline{V} + \overline{V} ~\subseteq~ W</math> 
(as will be shown, the set <math>C - C</math> is in fact a neighborhood of the origin in <math>X</math> because the topological interior of <math>C</math> in <math>X</math> is not empty). 
If <math>b \in B</math> then <math>H(b)</math> being bounded in <math>Y</math> implies that there exists some integer <math>n \in \N</math> such that <math>H(b) \subseteq n V</math> so if <math>h \in H,</math> then <math>b ~\in~ h^{-1}\left(n V\right) ~=~ n h^{-1}(V).</math> 
Since <math>h \in H</math> was arbitrary, 
<math display=block>b ~\in~ \bigcap_{h \in H} nh^{-1}(V) ~=~ n \bigcap_{h \in H} h^{-1}(V) ~\subseteq~ n C.</math> 
This proves that
<math display=block>B ~\subseteq~ \bigcup_{n \in \N} n C.</math>
Because <math>B</math> is of the second category in <math>X,</math> the same must be true of at least one of the sets <math>n C</math> for some <math>n \in \N.</math> 
The map <math>X \to X</math> defined by <math display="inline">x \mapsto \frac{1}{n} x</math> is a ([[Surjective function|surjective]]) [[Homeomorphism|homeomorphism]], so the set <math display="inline">\frac{1}{n} (n C) = C</math> is necessarily of the second category in <math>X.</math> 
Because <math>C</math> is closed and of the second category in <math>X,</math> its topological interior in <math>X</math> is not empty. 
Pick <math>c \in \operatorname{Int}_X C.</math> 
Because the map <math>X \to X</math> defined by <math>x \mapsto c - x</math> is a homeomorphism, the set 
<math display=block>N ~:=~ c - \operatorname{Int}_X C ~=~ \operatorname{Int}_X (c - C)</math> 
is a neighborhood of <math>0 = c - c</math> in <math>X,</math> which implies that the same is true of its superset <math>C - C.</math> 
And so for every <math>h \in H,</math> 
<math display=block>h(N) ~\subseteq~ h(c - C) ~=~ h(c) - h(C) ~\subseteq~ \overline{V} - \overline{V} ~\subseteq~ W.</math> 
This proves that <math>H</math> is equicontinuous. 
Q.E.D.

{{hr|1}}

{{em|Proof that <math>B = X</math>:}}

Because <math>H</math> is equicontinuous, if <math>S \subseteq X</math> is bounded in <math>X</math> then <math>H(S)</math> is uniformly bounded in <math>Y.</math> 
In particular, for any <math>x \in X,</math> because <math>S := \{x\}</math> is a bounded subset of <math>X,</math> <math>H(\{x\}) = H(x)</math> is a uniformly bounded subset of <math>Y.</math> Thus <math>B = X.</math> 
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. 

{{math theorem | name = Theorem{{sfn|Rudin|1991|pp=45−46}} | math_statement=
Suppose that <math>h_1, h_2, \ldots</math> is a sequence of continuous linear maps between two [[Topological vector space|topological vector space]]s <math>X</math> and <math>Y.</math> 

# If the set <math>C</math> of all <math>x \in X</math> for which <math>h_1(x), h_2(x), \ldots</math> is a Cauchy sequence in <math>Y</math> is of the second category in <math>X,</math> then <math>C = X.</math> 
# If the set <math>L</math> of all <math>x \in X</math> at which the limit <math>h(x) := \lim_{n \to \infty} h_n(x)</math> exists in <math>Y</math> is of the second category in <math>X</math> and if <math>Y</math> is a [[Complete topological vector space|complete]] [[Metrizable topological vector space|metrizable topological vector space]] (such as a [[Fréchet space]] or an [[F-space]]), then <math>L = X</math> and <math>h : X \to Y</math> is a continuous linear map. 
}}

{{math theorem | name = Theorem{{sfn|Rudin|1991|p=46}} | math_statement=
If <math>h_1, h_2, \ldots</math> is a sequence of continuous linear maps from an [[F-space]] <math>X</math> into a Hausdorff topological vector space <math>Y</math> such that for every <math>x \in X,</math> the limit
<math display=block>h(x) ~:=~ \lim_{n \to \infty} h_n(x)</math> 
exists in <math>Y,</math> then <math>h : X  \to Y</math> is a continuous linear map and the maps <math>h, h_1, h_2, \ldots</math> are equicontinuous. 
}}

If in addition the domain is a [[Banach space]] and the codomain is a normed space then <math>\|h\| \leq \liminf_{n \to \infty} \left\|h_n\right\| < \infty.</math> 

====Complete metrizable domain====

{{harvtxt|Dieudonné|1970}} proves a weaker form of this theorem with [[Fréchet space]]s rather than the usual Banach spaces.

{{math theorem| name = Theorem{{sfn|Rudin|1991|pp=42−47}} | math_statement=
Let <math>H \subseteq L(X, Y)</math> be a set of continuous linear operators from a [[Complete topological vector space|complete]] [[Metrizable topological vector space|metrizable topological vector space]] <math>X</math> (such as a [[Fréchet space]] or an [[F-space]]) into a [[Hausdorff space|Hausdorff]] [[Topological vector space|topological vector space]] <math>Y.</math> 
If for every <math>x \in X,</math> the orbit
<math display=block>H(x) := \{h(x) : h \in H\}</math>
is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y</math> then <math>H</math> is equicontinuous.

So in particular, if <math>Y</math> is also a normed space and if
<math display=block>\sup_{h \in H} \|h(x)\| < \infty \quad \text{ for every } x \in X,</math> 
then <math>H</math> is equicontinuous.
}}

==See also==

* {{annotated link|Barrelled space}}
* {{annotated link|Ursescu theorem}}

==Notes==

{{reflist|group=note}}
{{reflist|group=proof}}

==Citations==

{{reflist}}

==Bibliography==

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* {{citation| last=Sokal| first=Alan|  title=A really simple elementary proof of the uniform boundedness theorem| journal=Amer. Math. Monthly| volume=118| pages=450–452| year=2011| issue=5| arxiv=1005.1585| doi=10.4169/amer.math.monthly.118.05.450| s2cid=41853641}}.
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

{{Functional analysis}}

[[Category:Articles containing proofs]]
[[Category:Functional analysis]]
[[Category:Mathematical principles]]
[[Category:Theorems in functional analysis]]

{{Sourceattribution|Uniform boundedness principle}}