Editing Functional analysis (section)

===Uniform boundedness principle===
{{main|Banach-Steinhaus theorem}}
The [[uniform boundedness principle]] or [[Banach–Steinhaus theorem]] is one of the fundamental results in functional analysis. Together with the [[Hahn–Banach theorem]] and the [[open mapping theorem (functional analysis)|open mapping theorem]], it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of [[continuous linear operator]]s (and thus bounded operators) whose domain is a [[Banach space]], pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by [[Stefan Banach]] and [[Hugo Steinhaus]] but it was also proven independently by [[Hans Hahn (mathematician)|Hans Hahn]].

{{math theorem | name = Theorem (Uniform Boundedness Principle) | math_statement = Let <math>X</math> be a [[Banach space]] and <math>Y</math> be a [[normed vector space]]. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y</math>. If for all <math>x</math> in <math>X</math> one has
<math display="block">\sup\nolimits_{T \in F} \|T(x)\|_Y  < \infty, </math>
then
<math display="block">\sup\nolimits_{T \in F} \|T\|_{B(X,Y)}  < \infty.</math>}}