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Banach Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
The second fundamental theorem is the open mapping theorem (Theorem 3.5.3 for closed linear surjective maps between Fréchet spaces). It asserts that a continuous surjective linear map A between Banach spaces X and Y is open, that is, the image A(G) of any open set in X is open in Y. The closed graph theorem (Theorem 3.6.1 for closed linear maps between Fréchet spaces) is a corollary of the open mapping theorem, and asserts that a linear map between two Banach spaces with a closed graph is continuous. One version of the open mapping theorem and the closed graph theorem was proved by Banach in 1929. A more general version of these theorems was proved by Schauder in 1930.
Closed Graph Theorem and Open Mapping Theorem
Published in P.N. Natarajan, Functional Analysis and Summability, 2020
The “open mapping theorem” now follows: Open mapping theoremIf X, Y are Banach spaces and A is a continuous linear transformation of X onto Y, then A is an open mapping. ProofThis follows immediately from Theorem 65, i.e., the “interior mapping principle,” by taking X in the place of D.A one-to-one continuous linear mapping of a Banach space X onto a Banach space Y, over the same field, is a linear homeomorphism of X onto Y. ProofLet A be a continuous one-to-one linear mapping of X onto Y. Then, by the open mapping theorem, A(U) is an open set of Y whenever U is an open set in X. Since A is one-to-one, (A−1)−1(U) = A(U), i.e., (A−1)−1(U) is open whenever U is open, i.e., A−1 is continuous. Thus A is a linear homeomorphism of X onto Y, completing the proof.
The algebra of bounded operators on a Banach space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
It is also true in the setting of Banach spaces that a surjective and bounded linear map is an open map — this is the open mapping theorem. We challenge the reader to try to prove this general result (the proof does not require learning new material, only a bit of ingenuity).
An open mapping theorem for nonlinear operator equations associated with elliptic complexes
Published in Applicable Analysis, 2023
In contrast to [6] we consider this problem in special Sobolev–Bochner type spaces, cf. [9] for the de Rham complex on the torus in the case where i = 1. Namely, using the standard tools, such as the interpolation Gagliardo–Nirenberg inequalities, Gronwall type lemmas and Faedo–Galerkin approximations, see, for instance, [1,2], we show that suitable linearizations of our problem have one and only one solution and nonlinear problem (4) has at most one solution in the constructed spaces. Applying the implicit function theorem for Banach spaces we prove that the image of is open in selected spaces (thus, obtaining the so-called ‘open mapping theorem’). However, we do not discuss here a much more delicate question on the existence of solution to nonlinear problem (4) aiming at the maximal generality of the nonlinear term for the open mapping theorem to be true. It is worth to note that even for the existence of weak ( distributional) solutions to (4) one has to impose rather restrictive but reasonable additional assumptions on the nonlinearity .