Main Page | See live article | Alphabetical index

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:

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.