Editing Uniform boundedness principle (section)

===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>
}}