Editing Uniform boundedness principle (section)

===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.
}}