space of analytic functions

space of analytic functions

For what follows suppose that

G

⊂

ℂ

is a region. We wish to take the set of all

holomorphic functions

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 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

⊂

G

such that

K

n

⊂

K

n

+

1

∘

(

interior

of

K

n

+

1

), such that

⋃

K

n

∘

=

G

and
such that if

K

is any compact subset of

G

, then

K

⊂

K

n

for some

n

.
Now define the quantity

ρ

n

⁢

(

f

,

g

)

for

f

,

g

∈

𝒪

⁢

(

G

)

as

ρ

n

⁢

(

f

,

g

)

:=

sup

z

∈

K

n

⁡

{

|

f

⁢

(

z

)

-

g

⁢

(

z

)

|

}

.

We define the metric on

𝒪

⁢

(

G

)

as

d

⁢

(

f

,

g

)

:=

∑

n

=

1

∞

(

1

2

)

n

⁢

ρ

n

⁢

(

f

,

g

)

1

+

ρ

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

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

ρ

n

⁢

(

f

,

g

)

with

ρ

n

⁢

(

f

,

g

)

:=

sup

z

∈

K

n

⁡

{

σ

⁢

(

f

⁢

(

z

)

,

g

⁢

(

z

)

)

}

,

where

σ

is either the

spherical metric

on the Riemann sphere, or alternatively the metric induced by

embedding

the Riemann sphere in

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

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

)

=

∞

and
that is not a function in

M

⁢

(

G

)

.

Remark

.

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

continuous

on

G

¯

(

closure

of

G

).

Remark

.

We can similarly define the space of

continuous functions

, and treat

𝒪

⁢

(

G

)

and

M

⁢

(

G

)

as

subspaces

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

^

)

.

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

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