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 : 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:

• 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.
• If A : X → Y is a linear operator between the Banach spaces X and Y, and if for every sequence (x_n) in X with x_n → 0 and Ax_n → 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(U) is an open subset of C.