Editing Functional analysis (section)

===Open mapping theorem===
{{main|Open mapping theorem (functional analysis)}}
The [[open mapping theorem (functional analysis)|open mapping theorem]], also known as the Banach–Schauder theorem (named after [[Stefan Banach]] and [[Juliusz Schauder]]), is a fundamental result which states that if a [[Bounded linear operator|continuous linear operator]] between [[Banach space]]s is [[surjective]] then it is an [[open map]]. More precisely,<ref name=rudin/>
{{math theorem | name = Open mapping theorem | math_statement = If <math>X</math> and <math>Y</math> are Banach spaces and <math>A:X\to Y</math> is a surjective continuous linear operator, then <math>A</math> is an open map (that is, if <math>U</math> is an [[open set]] in <math>X</math>, then <math>A(U)</math> is open in <math>Y</math>).}}

The proof uses the [[Baire category theorem]], and completeness of both <math>X</math> and <math>Y</math> is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a [[normed space]], but is true if <math>X</math> and <math>Y</math> are taken to be [[Fréchet space]]s.