# Open mapping theorem

In

mathematics, there are two theorems with the name

**open mapping theorem**.

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**.