Open mapping theorem (functional analysis)

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 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
$\exists k \in \N: \qquad \left(\overline \right)^ \neq \varnothing.$

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 $\left(x_n\right)$ 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 (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. 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

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References

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{{Topological vector spaces

Articles containing proofs
Theorems in functional analysis