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In complex analysis, the open mapping theorem states that if ''U'' is a domain 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, 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 ''f''(''x'') = ''x''2 is not an open map, as the image of the open interval (−1, 1) is the half-open interval [0, 1). The theorem for example implies that a non-constant holomorphic function cannot map an open disk ''onto'' a portion of any 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== Assume ''f'' : ''U'' → C is a non-constant holomorphic function and ''U'' is a domain of the complex plane. We have to show that every point in ''f''(''U'') is an interior point of ''f''(''U''), i.e. that every point in ''f''(''U'') has a neighborhood (open disk) which is also in ''f''(''U''). Consider an arbitrary ''w''0 in ''f''(''U''). Then there exists a point ''z''0 in ''U'' such that ''w''0 = ''f''(''z''0). Since ''U'' is open, we can find ''d'' > 0 such that the closed disk ''B'' around ''z''0 with radius ''d'' is fully contained in ''U''. Consider the function ''g''(''z'') = ''f''(''z'')−''w''0. Note that ''z''0 is a root of the function. We know that ''g''(''z'') is not constant and holomorphic. The roots of ''g'' are isolated by the identity theorem, 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, on which |''g''(''z'')| is a positive continuous function, so the extreme value theorem guarantees the existence of a positive minimum ''e'', that is, ''e'' is the minimum of |''g''(''z'')| for ''z'' on the boundary of ''B'' and ''e'' > 0. Denote by ''D'' the open disk around ''w''0 with radius ''e''. By Rouché's theorem, the function ''g''(''z'') = ''f''(''z'')−''w''0 will have the same number of roots (counted with multiplicity) in ''B'' as ''h''(''z''):=''f''(''z'')−''w1'' for any ''w1'' in ''D''. This is because ''h''(''z'') = ''g''(''z'') + (''w''0 - ''w''1), and for ''z'' on the boundary of ''B'', |''g''(''z'')| ≥ ''e'' > |''w''0 - ''w''1|. Thus, for every ''w''1 in ''D'', there exists at least one ''z''1 in ''B'' such that ''f''(''z''1) = ''w1''. This means that the disk ''D'' is contained in ''f''(''B''). The image of the ball ''B'', ''f''(''B'') is a subset of the image of ''U'', ''f''(''U''). Thus ''w''0 is an interior point of ''f''(''U''). Since ''w''0 was arbitrary in ''f''(''U'') we know that ''f''(''U'') is open. Since ''U'' was arbitrary, the function ''f'' is open. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Open mapping theorem (complex analysis)」の詳細全文を読む スポンサード リンク
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