space of analytic functions
For what follows suppose that is a region. We wish to take the set of all holomorphic functions on , denoted by , and make it into a metric space. We will define a metric such that convergence in this metric is the same as uniform convergence on compact subsets of . We will call this the space of analytic functions on .
We define the metric on as
This can be shown to be a metric. Furthermore, it can be shown that the topology generated by this metric is independent of the choice of , even though the actual values of the metric do depend on the particular we have chosen. Finally, it can be shown that convergence in is the same as uniform convergence on compact subsets. It is known that if you have a sequence of analytic functions on that converge uniformly on compact subsets, then the limit is in fact analytic in , and thus is a complete space.
Similarly, we can treat the functions that are meromorphic on , and define to be the space of meromorphic functions on . We assume that the functions take the value at their poles, so that they are defined at every point of . That is, they take their values in the Riemann sphere, or the extended complex plane. We just need to replace the definition of with
where is either the spherical metric on the Riemann sphere, or alternatively the metric induced by embedding the Riemann sphere in . Both of those metrics produce the same topology, and that is all that we care about. The rest of the definition is the same as that of . There is, however, one small glitch here. is not a complete metric space. It is possible that functions in go off to infinity pointwise, but this is the worst that can happen. For example, the sequence is a sequence of meromorphic functions on , and this sequence is Cauchy in , but the limit would be and that is not a function in .
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
|Title||space of analytic functions|
|Date of creation||2013-03-22 14:24:48|
|Last modified on||2013-03-22 14:24:48|
|Last modified by||jirka (4157)|
|Synonym||space of holomorphic functions|
|Defines||space of meromorphic functions|