Editing Uniform boundedness principle (section)

===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}}
}}