In functional analysis, the open mapping theorem, also known as the BanachSchauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, and 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:
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 nonconstant holomorphic function, then f(U) is an open subset of C.
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