Open mapping theorem

In mathematics, there are two theorems with the name "open mapping theorem".

Functional analysis

In functional analysis, the open mapping theorem, also known as the Banach-Schauder theorem, is a fundamental result which states: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

The proof uses the Baire category theorem.

The open mapping theorem has two important consequences:

• If A : X → Y is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A^-1 : Y → X is continuous as well (this is called the bounded inverse theorem).
• If A : X → Y is a linear operator between the Banach spaces X and Y, and if for every sequence (x_n) in X with x_n → 0 and Ax_n → y it follows that y = 0, then A is continuous (Closed graph theorem).

Complex analysis

In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subsets of U to open subsets of C).

The theorem for example implies that a non-constant holomorphic function cannot map an open disk onto a portion of a line.

Proof

Blue dots represent zeros. Black spikes represent poles. The boundary of an open set is given by a dashed line. Note that all poles are exterior to the open set.

First assume $f$ is a non-constant holomorphic function and $U$ is a connected open subset of the complex plane. If every point in $f(U)$ is an interior point of $f(U)$ then $f(U)$ is open. Thus, if every point in $f(U)$ is contained in a disk which is contained in $f(U)$, then $f(U)$ is open.

Around every point in $U$, there is a relevant ball in $U$. Consider an arbitrary $z_{0}$ in $U$, and then consider its image point, $w_{0}=f(z_{0})$. Then $f(z_{0})-w_{0}=0$, making $z_{0}$ a root of $f(z)-w_{0}$. The function $f(z)-w_{0}$ may have another root at a distance $d_{1}$ from $z_{0}$. Additionally, the distance from $z_{0}$ to a point not in $U$ shall be written $d_{2}$. Any ball $B$ of radius less than the minimum of $d_{1}$ and $d_{2}$ will be contained in $U$, and at least one exists because $d_{1},d_{2}>0$.

Denote by $B_{2}$ the ball around $w_{0}$ with radius $e$ whose elements are written $w$. By Rouché's theorem or the Argument principle, the function $f(z)-w_{0}$ will have the same number of roots as $f(z)-w$ for any $w$ within a distance $e$ of $f(z_{0})$. Let $z_{1}$ be the root, or one of the roots of $f(z)-w$ just shown to exist. Thus, for every $w$ in $B_{2}$, there exists a $z_{1}$ in $B$ so that $f(z_{1})=w$, The image of B_2 is a subset of the image of B, which is a subset of $f(U)$.

Thus $w$ is an interior point of $f(U)$ for arbitrary $w$, and the theorem is proved.