Open mapping theorem (complex analysis): Difference between revisions

Revision as of 15:51, 24 November 2008

Revision as of 15:52, 24 November 2008

Line 13:

Consider an arbitrary <math>z_0</math> in <math>U</math>. Since ''U'' is open, we can find <math>d>0</math> such that the closed disk <math>B</math> around ''z''₀ 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''. Consider the [[Image (mathematics) | image]] point, <math>w_0 = f(z_0)</math>. Then <math>f(z_0)-w_0 = 0</math>, making <math>z_0</math> a [[Root (mathematics) | root]] of the function <math>g(z)=f(z)-w_0</math>.

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>.

== References ==