Open mapping theorem

From Academic Kids

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, 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 : 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 (this is called the inverse mapping theorem).
  • 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 is an open map (i.e. it sends open subsets of U to open subsets of C).es:Teorema de la función abierta he:משפט ההעתקה הפתוחה

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