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Banach's theorems

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The statement of the theorem

Here are the main general results about Banach spaces that go back to the time of Banach's book (Banach (1932)) and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional. Banach–Steinhaus Theorem—Let XX be a Banach space and YY be a normed vector space. Suppose that FF is a collection of continuous linear operators from XX to Y.Y. The uniform boundedness principle states that if for all xx in XX we have sup⁡T∈F∥T(x)∥Y<∞,\sup _{T\in F}\|T(x)\|_{Y}<\infty , then sup⁡T∈F∥T∥Y<∞.\sup _{T\in F}\|T\|_{Y}<\infty . The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where XX is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood UU of 0\mathbf {0} in XX such that all TT in FF are uniformly bounded on U,U, sup⁡T∈Fsup⁡x∈U  ∥T(x)∥Y<∞.\sup _{T\in F}\sup _{x\in U}\;\|T(x)\|_{Y}<\infty . The Open Mapping Theorem—Let XX and YY be Banach spaces and T:X→YT:X\to Y be a surjective continuous linear operator, then TT is an open map. Corollary—Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism. The First Isomorphism Theorem for Banach spaces—Suppose that XX and YY are Banach spaces and that T∈B(X,Y).T\in B(X,Y). Suppose further that the range of TT is closed in Y.Y. Then X/ker⁡TX/\ker T is isomorphic to T(X).T(X). This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps. Corollary—If a Banach space XX is the internal direct sum of closed subspaces M1,…,Mn,M_{1},\ldots ,M_{n}, then XX is isomorphic to M1⊕⋯⊕Mn.M_{1}\oplus \cdots \oplus M_{n}. This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M1⊕⋯⊕MnM_{1}\oplus \cdots \oplus M_{n} onto XX sending m1,⋯ ,mnm_{1},\cdots ,m_{n} to the sum m1+⋯+mn.m_{1}+\cdots +m_{n}. The Closed Graph Theorem—Let T:X→YT:X\to Y be a linear mapping between Banach spaces. The graph of TT is closed in X×YX\times Y if and only if TT is continuous.
Source: Wikipedia