Editing Banach space (section)

===Banach's theorems===

Here are the main general results about Banach spaces that go back to the time of Banach's book ({{harvtxt|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 [[Interior (topology)|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.

{{math theorem|name=[[Uniform boundedness principle|Banach–Steinhaus Theorem]]|math_statement=Let <math>X</math> be a Banach space and <math>Y</math> be a [[normed vector space]]. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y.</math> The uniform boundedness principle states that if for all <math>x</math> in <math>X</math> we have <math>\sup_{T \in F} \|T(x)\|_Y < \infty,</math> then <math>\sup_{T \in F} \|T\|_Y < \infty.</math>}}

The Banach–Steinhaus theorem is not limited to Banach spaces. 
It can be extended for example to the case where <math>X</math> is a [[Fréchet space]], provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood <math>U</math> of <math>\mathbf{0}</math> in <math>X</math> such that all <math>T</math> in <math>F</math> are uniformly bounded on <math>U,</math>
<math display=block>\sup_{T \in F} \sup_{x \in U} \; \|T(x)\|_Y < \infty.</math>

{{math theorem|name=[[Open mapping theorem (functional analysis)|The Open Mapping Theorem]]|math_statement=Let <math>X</math> and <math>Y</math> be Banach spaces and <math>T : X \to Y</math> be a surjective continuous linear operator, then <math>T</math> is an open map.}}

{{math theorem|name=Corollary | math_statement = Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.}}

{{math theorem|name=The First Isomorphism Theorem for Banach spaces | math_statement= Suppose that <math>X</math> and <math>Y</math> are Banach spaces and that <math>T \in B(X, Y).</math> Suppose further that the range of <math>T</math> is closed in <math>Y.</math> Then <math>X / \ker T</math> is isomorphic to <math>T(X).</math>}}

This result is a direct consequence of the preceding ''Banach isomorphism theorem'' and of the canonical factorization of bounded linear maps.

{{math theorem|name=Corollary|math_statement=If a Banach space <math>X</math> is the internal direct sum of closed subspaces <math>M_1, \ldots, M_n,</math> then <math>X</math> is isomorphic to <math>M_1 \oplus \cdots \oplus M_n.</math>}}

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from <math>M_1 \oplus \cdots \oplus M_n</math> onto <math>X</math> sending <math>m_1, \cdots, m_n</math> to the sum <math>m_1 + \cdots + m_n.</math>

{{math theorem|name=[[Closed graph theorem|The Closed Graph Theorem]]|math_statement= Let <math>T : X \to Y</math> be a linear mapping between Banach spaces. The graph of <math>T</math> is closed in <math>X \times Y</math> if and only if <math>T</math> is continuous.}}