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Open mapping theorem

In mathematics, there are two theorems with the name open mapping theorem.

Functional analysis

In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : XY 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:

  • If A : XY is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A-1 : YX is continuous as well.
  • If A : XY is a linear operator between the Banach spaces X and Y, and if for every sequence (xn) in X with xn → 0 and Axny it follows that y = 0, then A is continuous (Closed graph theorem[?]).

Complex analysis

In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : UC is a non-constant holomorphic function, then f(U) is an open subset of C.

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