# space of analytic functions

For what follows suppose that $G\subset \u2102$ is a region. We wish to take the set of all holomorphic functions^{} on $G$, denoted by $\mathcal{O}(G)$, and make it into a metric space. We will define a metric such that convergence in this metric is the same as uniform convergence^{} 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 ${K}_{n}\subset G$ such that ${K}_{n}\subset {K}_{n+1}^{\circ}$ (interior of ${K}_{n+1}$), such that $\bigcup {K}_{n}^{\circ}=G$ and such that if $K$ is any compact subset of $G$, then $K\subset {K}_{n}$ for some $n$. Now define the quantity ${\rho}_{n}(f,g)$ for $f,g\in \mathcal{O}(G)$ as

$${\rho}_{n}(f,g):=\underset{z\in {K}_{n}}{sup}\{|f(z)-g(z)|\}.$$ |

We define the metric on $\mathcal{O}(G)$ as

$$d(f,g):=\sum _{n=1}^{\mathrm{\infty}}{\left(\frac{1}{2}\right)}^{n}\frac{{\rho}_{n}(f,g)}{1+{\rho}_{n}(f,g)}.$$ |

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 ${K}_{n}$, even though
the actual values of the metric do depend on the particular ${K}_{n}$ 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 functions^{} on $G$ that converge uniformly on compact subsets, then the limit is in fact analytic in $G$, and thus $\mathcal{O}(G)$ 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 $\mathrm{\infty}$ 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 ${\rho}_{n}(f,g)$ with

$${\rho}_{n}(f,g):=\underset{z\in {K}_{n}}{sup}\{\sigma (f(z),g(z))\},$$ |

where $\sigma $ is either the spherical metric on the Riemann sphere, or alternatively the metric induced by embedding the Riemann sphere in ${\mathbb{R}}^{3}$. 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 $\mathcal{O}(G)$. 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 ${f}_{n}(z)=n$ is a sequence of meromorphic functions on $G$, and this sequence is Cauchy in $M(G)$, but the limit would be $f(z)=\mathrm{\infty}$ and that is not a function in $M(G)$.

###### Remark.

Note that $\mathcal{O}(G)$ is sometimes denoted by $H(G)$ in literature. Also note that $A(G)$ is usually reserved for functions which are
analytic on $G$ and continuous^{} on $\overline{G}$ (closure^{} of $G$).

###### Remark.

We can similarly define the space of continuous functions, and treat $\mathcal{O}(G)$ and $M(G)$ as subspaces^{} of that. That is, $\mathcal{O}(G)$
would be a subspace of $C(G,\u2102)$ and $M(G)$ would be a subspace
of $C(G,\widehat{\u2102})$.

## References

- 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 |