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* : *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).

## 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* : *U* → **C** is a non-constant holomorphic function, then *f*(*U*) is an open subset of **C**.