Editing Uniform boundedness principle (section)

====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.
}}