space of analytic functions

For what follows suppose that G⊂ℂ is a region. We wish to take the set of all holomorphic functionsMathworldPlanetmath on G, denoted by 𝒪⁢(G), and make it into a metric space. We will define a metric such that convergence in this metric is the same as uniform convergenceMathworldPlanetmath on compact subsets of G. We will call this the space of analytic functions on G.

It is known that there exists a sequence of compact subsets Kn⊂G such that Kn⊂Kn+1∘ (interior of Kn+1), such that ⋃Kn∘=G and such that if K is any compact subset of G, then K⊂Kn for some n. Now define the quantity ρn⁢(f,g) for f,g∈𝒪⁢(G) as


We define the metric on 𝒪⁢(G) as


This can be shown to be a metric. Furthermore, it can be shown that the topologyMathworldPlanetmathPlanetmath generated by this metric is independent of the choice of Kn, even though the actual values of the metric do depend on the particular Kn we have chosen. Finally, it can be shown that convergence in d is the same as uniform convergence on compact subsets. It is known that if you have a sequence of analytic functionsMathworldPlanetmath on G that converge uniformly on compact subsets, then the limit is in fact analytic in G, and thus 𝒪⁢(G) is a complete space.

Similarly, we can treat the functions that are meromorphic on G, and define M⁢(G) to be the space of meromorphic functions on G. We assume that the functions take the value ∞ at their poles, so that they are defined at every point of G. That is, they take their values in the Riemann sphere, or the extended complex plane. We just need to replace the definition of ρn⁢(f,g) with


where σ is either the spherical metric on the Riemann sphere, or alternatively the metric induced by embedding the Riemann sphere in ℝ3. 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 𝒪⁢(G). There is, however, one small glitch here. M⁢(G) is not a complete metric space. It is possible that functions in M⁢(G) go off to infinity pointwise, but this is the worst that can happen. For example, the sequence fn⁢(z)=n is a sequence of meromorphic functions on G, and this sequence is Cauchy in M⁢(G), but the limit would be f⁢(z)=∞ and that is not a function in M⁢(G).


Note that 𝒪⁢(G) is sometimes denoted by H⁢(G) in literature. Also note that A⁢(G) is usually reserved for functions which are analytic on G and continuousMathworldPlanetmath on G¯ (closurePlanetmathPlanetmath of G).


We can similarly define the space of continuous functions, and treat 𝒪⁢(G) and M⁢(G) as subspacesMathworldPlanetmath of that. That is, 𝒪⁢(G) would be a subspace of C⁢(G,ℂ) and M⁢(G) would be a subspace of C⁢(G,ℂ^).


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title space of analytic functions
Canonical name SpaceOfAnalyticFunctions
Date of creation 2013-03-22 14:24:48
Last modified on 2013-03-22 14:24:48
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32A10
Classification msc 30D20
Synonym space of holomorphic functions
Related topic Holomorphic
Related topic Meromorphic
Related topic MontelsTheorem
Defines space of meromorphic functions