Editing Uniform boundedness principle (section)

==Generalizations==

In a [[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]]s on which the uniform boundedness principle holds eventually led to [[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 mapping]]s 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===

{{Main|Uniformly bounded sets (topological vector space)}}

A [[Family of sets|family]] <math>\mathcal{B}</math> of subsets of a [[topological vector space]] <math>Y</math> is said to be {{em|[[Uniformly bounded sets (topological vector space)|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]]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]] <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 (topology)|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]]s between two  [[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 (group theory)|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]], 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]]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]] (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]] <math>X</math> (such as a [[Fréchet space]] or an [[F-space]]) into a [[Hausdorff space|Hausdorff]] [[topological vector space]] <math>Y.</math> 
If for every <math>x \in X,</math> the [[Orbit (group theory)|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.
}}