Open mapping theorem

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

[edit] 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 inverse mapping 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).

[edit] 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.

[edit] 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 z₀ in U, and then consider its image point, w₀ = f(z₀). Then f(z₀) − w₀ = 0, making z₀ a root of f(z) − w₀. The function f(z) − w₀ may have another root at a distance d₁ from z₀. Additionally, the distance from z₀ to a point not in U shall be written d₂. Any ball B of radius less than the minimum of d₁ and d₂ will be contained in U, and at least one exists because d₁,d₂ > 0.

Denote by B₂ the ball around w₀ with radius e whose elements are written w. By Rouché's theorem or the Argument principle, the function f(z) − w₀ will have the same number of roots as f(z) − w for any w within a distance e of f(z₀). Let z₁ be the root, or one of the roots of f(z) − w just shown to exist. Thus, for every w in B₂, there exists a z₁ in B so that f(z₁) = 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.