Functional analysis: Difference between revisions

Revision as of 23:47, 16 October 2022

Revision as of 23:48, 16 October 2022

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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/>

: '''Open mapping theorem.''' 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.

===Closed graph theorem===