Proof of the Open Mapping Theorem
The Open Mapping Theorem states:
Letbe a function analytic and non – constant on a regionand letbe an open subset ofThenis open.
To prove thatis open we need to show that ifthen there exists such that
Sincethere existssuch thatFurther, the solutions of the equationare isolated sinceis non constant and analytic and so we can find an open discinwith centreand radius sufficiently small such thatforThus, ifis a circle inwith centrethen the imageis a closed contour which does not pass through
is compact, being the continuous image of a compact set, so the complement ofis open and we can chooseso that lies in the complement of
The winding number ofabout each point of the disc is equal to
Nowby the Argument Principle sinceso for
Thus by the Argument Principle again, the equationhas at least one solution inside for eachsuch thathence as required.