Open Mapping Theorem (Functional Analysis)
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 the...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 Fréchet spaces.
