information:extraction
Open mapping theorem

The open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. More precisely,

Open mapping theorem — If X {\displaystyle X} and Y {\displaystyle Y} are Banach spaces and A: X → Y {\displaystyle A:X\to Y} is a surjective continuous linear operator, then A {\displaystyle A} is an open map (that is, if U {\displaystyle U} is an open set in X {\displaystyle X}, then A (U) {\displaystyle A(U)} is open in Y {\displaystyle Y}).

The proof uses the Baire category theorem, and completeness of both X {\displaystyle X} and Y {\displaystyle Y} is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if X {\displaystyle X} and Y {\displaystyle Y} are taken to be Fréchet spaces.