Open mapping theorem (complex analysis): Difference between revisions

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{{Short description|Theorem on holomorphic functions}}

In [[complex analysis]], the '''open mapping theorem''' states that if ''<math>U''</math> is a [[connectedDomain space(mathematical analysis)|connected___domain]] open subset of the [[complex plane]] '''<math>\mathbb{C'''}</math> and ''<math>f'' : ''U'' →\to '''\mathbb{C'''}</math> is a non-constant [[holomorphic]] function]], then ''<math>f''</math> is an [[open map]] (i.e. it sends open subsets of ''<math>U''</math> to open subsets of '''<math>\mathbb{C'''}</math>, and we have [[invariance of ___domain]].).

The open mapping theorem points to the sharp difference between holomorphy and real-differentiability. On the [[real line]], for example, the differentiable function ''<math>f''(''x'') = ''x''<sup>^2</supmath> is not an open map, as the image of the [[open interval]] <math>(&minus;-1, 1)</math> is the half-open interval <math>[0, 1)</math>.

The theorem for example implies that a non-constant [[holomorphic function]] cannot map an open disk ''[[onto]]'' a portion of any real line embedded in the complex plane. Images of holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

==Proof==

[[Image: OpenMapping1Openmappingtheorem.png |framethumb|right|Blueupright=1.4| Black dots represent zeros of ''<math>g''(''z'')</math>. Black spikesannuli represent poles. The boundary of the open set ''<math>U''</math> is given by the dashed line. Note that all poles are exterior to the open set. The smaller red circledisk is the set ''<math>B''</math>, constructedcentered in theat proof<math>z_0</math>.]]

Assume ''<math>f'':'' U''\to &rarr; '''\mathbb{C'''}</math> is a non-constant holomorphic function and <math>U</math> is a connected [[openDomain subset(mathematical analysis)|___domain]] of the complex plane. We have to show that every [[Point (geometry) | point]] in <math>f(U)</math> is an [[interior point]] of <math>f(U)</math>, i.e. that every point in <math>f(U)</math> ishas containeda inneighborhood a(open disk) which is containedalso in <math>f(U)</math>.

Consider an arbitrary <math>w_0</math> in <math>f(U)</math>. Then there exists a point <math>z_0</math> in ''<math>U''</math> such that <math>w_0 = f(z_0)</math>. Since ''<math>U''</math> is open, we can find <math>d > 0</math> such that the closed disk <math>B</math> around <math>z_0</math> with radius ''<math>d''</math> is fully contained in ''<math>U''. Since ''U'' is connected and ''f'' is not constant on ''U'', we then know that ''f'' is not constant on ''B''</math>. Consider the function <math>g(z)=f(z)-w_0</math>. Note that <math>z_0</math> is a [[root of a function| root]] of the function.

We know that <math>g(z)</math> is non-constant and holomorphic. The roots of <math>g</math> are isolated by the [[identity theorem]], and by further decreasing the radius of the disk <math>B</math>, we can assure that <math>g(z)</math> has only a single root in <math>B</math> (although this single root may have multiplicity greater than 1).

We know that ''g''(''z'') is not constant, and by further decreasing ''d'', we can assure that ''g''(''z'') has only a single root in ''B''. (The roots of holomorphic non-constant functions are isolated.) Let ''e'' be the minimum of |''g''(''z'')| for ''z'' on the boundary of ''B'', a positive number. (The boundary of ''B'' is a circle and hence a [[compact set]], and |''g''(''z'')| is a [[continuous function]], so the [[extreme value theorem]] guarantees the existence of this minimum.) Denote by <math>D</math> the disk around <math>w_0</math> with [[radius]] <math>e</math>. By [[Rouché's theorem]], the function <math>g(z)=f(z)-w_0</math> will have the same number of roots in ''B'' as <math>f(z)-w</math> for any <math>w</math> within a distance <math>e</math> of <math>w_0</math>. Thus, for every <math>w</math> in <math>D</math>, there exists one (and only one) <math>z_1</math> in <math>B</math> so that <math>f(z_1) = w</math>. This means that the disk ''D'' is contained in ''f''(''B''), which is a subset of <math>f(U)</math>. Thus <math>w_0</math> is an [[interior point]] of <math>f(U)</math>.

The boundary of <math>B</math> is a circle and hence a [[compact set]], on which <math>|g(z)|</math> is a positive [[continuous function]], so the [[extreme value theorem]] guarantees the existence of a positive minimum <math>e</math>, that is, <math>e</math> is the minimum of <math>|g(z)|</math> for <math>z</math> on the boundary of <math>B</math> and <math>e>0</math>.

Denote by <math>D</math> the open disk around <math>w_0</math> with [[radius]] <math>e</math>. By [[Rouché's theorem]], the function <math>g(z)=f(z)-w_0</math> will have the same number of roots (counted with multiplicity) in <math>B</math> as <math>h(z) := f(z)-w_1</math> for any <math>w_1</math> in <math>D</math>. This is because

<math>h(z) = g(z) + (w_0-w_1)</math>, and for <math>z</math> on the boundary of <math>B</math>, <math>|g(z)| \geq e > |w_0-w_1|</math>. Thus, for every <math>w_1</math> in <math>D</math>, there exists at least one <math>z_1</math> in <math>B</math> such that <math>f(z_1) = w_1</math>. This means that the disk <math>D</math> is contained in <math>f(B)</math>.

The image of the ball <math>B</math>, <math>f(B)</math> is a subset of the image of <math>U</math>, <math>f(U)</math>. Thus <math>w_0</math> is an interior point of <math>f(U)</math>. Since <math>w_0</math> was arbitrary in <math>f(U)</math> we know that <math>f(U)</math> is open. Since <math>U</math> was arbitrary, the function <math>f</math> is open.

== Applications ==

*[[Maximum modulus principle]]

*[[Rouché's theorem]]

*[[Schwarz lemma]]

== See also ==

* [[Open mapping theorem (functional analysis)]]

== References ==

* {{citation|first=Walter|last=Rudin|authorlink=Walter Rudin|title=Real & Complex Analysis|publisher=McGraw-Hill|year=1966|isbn=0-07-054234-1}}

[[Category:ComplexTheorems in complex analysis]]

[[Category:Mathematical theorems]]

[[Category:Articles containing proofs]]

[[de:Offenheitssatz]]