Open Mapping Theorem (Functional Analysis) (Paperback)

High Quality Content by WIKIPEDIA articles! In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, 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, (Rudin 1973, Theorem 2.11): * If X and Y are Banach spaces and A: X Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y). The proof uses the Baire category theorem, and completeness of both X and 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 and Y are taken to be Frechet spaces.