Open mapping theorem (functional analysis) | Wikipedia audio article
2This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis)
00:01:11 1 Consequences
00:02:05 2 Proof
00:02:18 3 Generalizations
00:10:56 4 See also
00:13:29 5 References
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SUMMARY
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In functional analysis, the 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 continuous linear operator between Banach spaces is surjective then it is an open map. More precisely, (Rudin 1973, Theorem 2.11):
Open Mapping Theorem. If X and Y are Banach spaces and A : X → Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).One proof uses Baire's category theorem, and completeness of both X and Y 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 X and Y are taken to be Fréchet spaces.