Open mapping theorem (complex analysis): Difference between revisions

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Consider an arbitrary <math>w_0</math> in <math>f(U)</math>. Then there exists a point <math>z_0</math> in ''U'' such that <math>w_0 = f(z_0)</math>. Since ''U'' is open, we can find <math>d>0</math> such that the closed disk <math>B</math> around <math>z_0</math> with radius ''d'' is fully contained in ''U''. Since ''U'' is connected and ''f'' is not constant on ''U'', we then know that ''f'' is not constant on ''B'' because of [[Analytic continuation]]. 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 ''g''(''z'') is not constant and holomorphic. The reciprocal of any holomorphic ''g''(''z'') is [[meromorphic]] and has isolated poles. Thus the roots of holomorphic non-constant functions are isolated. In particular, the roots of ''g'' are isolated and by further decreasing the radius of the image disk ''d'', we can assure that ''g''(''z'') has only a single root in ''B'' (although this single root may have multiplicity greater than 1).

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 a minimum. Let ''e'' be the minimum of |''g''(''z'')| for ''z'' on the boundary of ''B'', a positive number.

Denote by <math>D</math> the disk around <math>w_0</math> with [[radius]] ''e''. 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 ''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 oneat (and onlyleast 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 <math>f(B)</math>.

The image of the ball ''B'', <math>f(B)</math> is a subset of the image of ''U'', <math>f(U)</math>. Thus <math>w_0</math> is an interior point of the image of an open set by a holomorphic function <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 ''U'' was arbitrary, the function <math>f</math> is open.