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Open mapping theorem
In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).
The proof uses the Baire category theorem.
The open mapping theorem has two important consequences:
- If A : X → Y is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A-1 : Y → X is continuous as well (this is called the inverse mapping theorem).
- If A : X → Y is a linear operator between the Banach spaces X and Y, and if for every sequence (xn) in X with xn → 0 and Axn → y it follows that y = 0, then A is continuous (Closed graph theorem).
In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C).
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