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In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a
bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...
or continuous linear operator between Banach spaces is surjective then it is an open map.


Classical (Banach space) form

This 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. Suppose A : X \to Y is a surjective continuous linear operator. In order to prove that A is an open map, it is sufficient to show that A maps the open unit ball in X to a neighborhood of the origin of Y. Let U = B_1^X(0), V = B_1^Y(0). Then X = \bigcup_ k U. Since A is surjective: Y = A(X) = A\left(\bigcup_ k U\right) = \bigcup_ A(kU). But Y is Banach so by
Baire's category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
\exists k \in \N: \qquad \left(\overline \right)^ \neq \varnothing. That is, we have c \in Y and r > 0 such that B_r(c) \subseteq \left(\overline \right)^\circ \subseteq \overline. Let v \in V, then c, c + r v \in B_r(c) \subseteq \overline. By continuity of addition and linearity, the difference r v satisfies r v \in \overline + \overline \subseteq \overline \subseteq \overline, and by linearity again, V \subseteq \overline where we have set L = 2 k / r. It follows that for all y \in Y and all \epsilon > 0, there exists some x \in X such that \qquad \, x\, _X \leq L \, y\, _Y \quad \text \quad \, y - A x\, _Y < \epsilon. \qquad (1) Our next goal is to show that V \subseteq A(2LU). Let y \in V. By (1), there is some x_1 with \left\, x_1\right\, < L and \left\, y - A x_1\right\, < 1/2. Define a sequence \left(x_n\right) inductively as follows. Assume: \, x_n\, < \frac \quad \text \quad \left\, y - A\left(x_1 + x_2 + \cdots + x_n\right)\right\, < \frac. \qquad (2) Then by (1) we can pick x_ so that: \, x_\, < \frac \quad \text \quad \left\, y - A\left(x_1 + x_2 + \cdots + x_n\right) - A\left(x_\right)\right\, < \frac, so (2) is satisfied for x_. Let s_n = x_1 + x_2 + \cdots + x_n. From the first inequality in (2), \left(s_n\right)is a Cauchy sequence, and since X is complete, s_n converges to some x \in X. By (2), the sequence A s_n tends to y and so Ax = y by continuity of A. Also, \, x\, = \lim_ \, s_n\, \leq \sum_^\infty \, x_n\, < 2 L. This shows that y belongs to A(2LU), so V \subseteq A(2LU) as claimed. Thus the image A(U) of the unit ball in X contains the open ball V / 2L of Y. Hence, A(U) is a neighborhood of the origin in Y, and this concludes the proof.


Related results


Consequences

The open mapping theorem has several important consequences: * If A : X \to Y is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A^ : Y \to X is continuous as well (this is called the bounded inverse theorem). * If A : X \to Y is a linear operator between the Banach spaces X and Y, and if for every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
\left(x_n\right) in X with x_n \to 0 and A x_n \to y it follows that y = 0, then A is continuous (the closed graph theorem).


Generalizations

Local convexity of X or Y  is not essential to the proof, but completeness is: the theorem remains true in the case when X and Y are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner: Furthermore, in this latter case if N is the kernel of A, then there is a canonical factorization of A in the form X \to X/N \overset Y where X / N is the
quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...
(also an F-space) of X by the closed subspace N. The quotient mapping X \to X / N is open, and the mapping \alpha is an isomorphism of topological vector spaces. The open mapping theorem can also be stated as Nearly/Almost open linear maps A linear map A : X \to Y between two topological vector spaces (TVSs) is called a (or sometimes, an ) if for every neighborhood U of the origin in the domain, the closure of its image \operatorname A(U) is a neighborhood of the origin in Y. Many authors use a different definition of "nearly/almost open map" that requires that the closure of A(U) be a neighborhood of the origin in A(X) rather than in Y, but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous. Every surjective linear map from locally convex TVS onto a barrelled TVS is
nearly open This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fund ...
. The same is true of every surjective linear map from a TVS onto a Baire TVS.


Consequences


Webbed spaces

Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.


See also

* * * * * * * * *


References


Bibliography

* * * * * * * * * * * * * * * * * * {{Topological vector spaces Articles containing proofs Theorems in functional analysis