# Montel's theorem

The Montel theorem (after Paul Montel ) is a theorem from function theory . He is concerned with the question of when a function sequence of holomorphic functions has a compact convergent subsequence . In this sense it is the analogue of the Bolzano-Weierstrass theorem for number sequences. It was found by Paul Montel in 1916.

## Statement of the sentence

The concept of the normal family introduced by Montel is fundamental for the formulation : A family of holomorphic functions is called normal if each sequence has a compact convergent subsequence. Here, convergence is considered with regard to the spherical metric, in particular convergence against is permitted. ${\ displaystyle {\ mathcal {F}}}$${\ displaystyle {\ mathcal {F}}}$${\ displaystyle \ infty}$

### Little phrase from Montel

A locally evenly bounded family of holomorphic functions is normal.

### Big set of Montel

Be a family of in an area and are holomorphic functions , . For everyone and apply . Then is normal. ${\ displaystyle {\ mathcal {F}}}$${\ displaystyle G}$${\ displaystyle a, b \ in \ mathbb {C}}$${\ displaystyle a \ neq b}$${\ displaystyle f \ in {\ mathcal {F}}}$${\ displaystyle z \ in G}$${\ displaystyle f (z) \ neq a, b}$${\ displaystyle {\ mathcal {F}}}$

Montel's little theorem follows directly from the big one. A comparatively simple proof of the large theorem can be found in an article by Lawrence Zalcman .

## Proof of Montel's Little Theorem

For the proof of Montel's little theorem one needs the following lemma:

### lemma

${\ displaystyle (f_ {n}) _ {n \ in \ mathbb {N}}}$be a function sequence that is holomorphic and locally uniformly bounded in one area . The crowd is close . ${\ displaystyle G}$${\ displaystyle P = \ {z \ in G: \ lim _ {n \ rightarrow \ infty} f_ {n} (z) \; {\ mbox {exists}} \}}$${\ displaystyle G}$

Then compact is convergent. ${\ displaystyle (f_ {n}) _ {n \ in \ mathbb {N}}}$

### Proof (lemma)

We want to show:

${\ displaystyle \ forall z_ {0} \ in G: \ \ exists r> 0: \ \ forall \ epsilon> 0: \ \ exists n_ {0}: \ left | f_ {n} (z) -f_ {m } (z) \ right | <\ epsilon \ quad \ forall z \ in B (z_ {0}, r), \ forall m, n> n_ {0},}$

where the open circular disk denotes the center and radius . ${\ displaystyle B (z_ {0}, r)}$${\ displaystyle z_ {0}}$${\ displaystyle r}$

Since the sequence of functions is locally evenly restricted, the following applies:

${\ displaystyle \ forall z_ {0} \ in G: \ \ exists R> 0, \ exists M> 0: \ left | f_ {n} (z) \ right | \ leq M \ quad \ forall z \ in B (z_ {0}, R), \ forall n \ in \ mathbb {N}.}$

Choose . ${\ displaystyle r = {\ frac {R} {2}}}$

Be now . Then ( Cauchy's integral formula ): ${\ displaystyle z, {\ tilde {z}} \ in B (z_ {0}, r)}$

{\ displaystyle {\ begin {aligned} \ left | f_ {n} (z) -f_ {n} ({\ tilde {z}}) \ right | & = \ left | {\ frac {1} {2 \ pi i}} \ oint _ {\ left | w-z_ {0} \ right | = R} {\ frac {f_ {n} (w)} {wz}} dw - {\ frac {1} {2 \ pi i}} \ oint _ {\ left | w-z_ {0} \ right | = R} {\ frac {f_ {n} (w)} {w - {\ tilde {z}}}} dw \ right | \\ & = \ left | {\ frac {z - {\ tilde {z}}} {2 \ pi i}} \ oint _ {\ left | w-z_ {0} \ right | = R} {\ frac {f_ {n} (w)} {(wz) (w - {\ tilde {z}})}} dw \ right | \ end {aligned}}}

Now one estimates the integral by the length of the curve and the maximum of the integrand (more precisely an estimate of the maximum):

${\ displaystyle \ left | {\ frac {z - {\ tilde {z}}} {2 \ pi i}} \ oint _ {\ left | w-z_ {0} \ right | = R} {\ frac { f_ {n} (w)} {(wz) (w - {\ tilde {z}})}} dw \ right | \ leq {\ frac {\ left | z - {\ tilde {z}} \ right | } {2 \ pi}} \ cdot 2 \ pi R \ cdot {\ frac {M} {r ^ {2}}} = R {\ frac {M} {r ^ {2}}} \ left | z- {\ tilde {z}} \ right | = 2 {\ frac {M} {r}} \ left | z - {\ tilde {z}} \ right |}$

So:

${\ displaystyle \ left | f_ {n} (z) -f_ {n} ({\ tilde {z}}) \ right | \ leq 2 {\ frac {M} {r}} \ left | z - {\ tilde {z}} \ right |}$

Now P lies close in G. So for every given ε one can choose finitely many from P so that the ε neighborhoods completely cover. (Since is compact, finitely many are sufficient.) Here we choose our ε in such a way that we then get exactly in combination with the above estimate . ${\ displaystyle p_ {i}}$${\ displaystyle B (z_ {0}, r)}$${\ displaystyle B (z_ {0}, r)}$${\ displaystyle \ epsilon / 3}$

${\ displaystyle \ exists p_ {1}, \ dots p_ {k} \ in B (z_ {0}, r): \ \ forall z \ in B (z_ {0}, r): \ \ exists a_ {j }: \ \ left | z-a_ {j} \ right | <{\ frac {\ epsilon} {3}} {\ frac {r} {(2M)}}}$

${\ displaystyle \ left | f_ {n} (z) -f_ {m} (z) \ right | \ leq \ left | f_ {n} (z) -f_ {n} (p_ {j}) \ right | + \ left | f_ {n} (p_ {j}) - f_ {m} (p_ {j}) \ right | + \ left | f_ {m} (p_ {j}) - f_ {m} (z) \ right | \ quad n, m> n_ {0}}$

${\ displaystyle p_ {j}}$be the closest to z . Then the first and last addend can be estimated using the two upper estimates. Since they converge on the pointwise, the mean term (for sufficiently large n) is also smaller than . ${\ displaystyle p_ {i}}$${\ displaystyle \ epsilon / 3}$${\ displaystyle f_ {n}}$${\ displaystyle a_ {j}}$${\ displaystyle \ epsilon / 3}$

So we get:

${\ displaystyle \ forall z \ in G: \ \ exists r> 0: \ \ forall \ epsilon> 0: \ exists n_ {0}: \ left | f_ {n} (z) -f_ {m} (z) \ right | \ leq \ epsilon \ quad \ forall z \ in B (z_ {0}, r), \ forall m, n> n_ {0}}$

### Proof (Montel's Theorem)

In order to be able to use the upper lemma, we first choose a countable, dense subset of the area . (e.g .: only those with rational real and imaginary parts) ${\ displaystyle \ {p_ {1}, p_ {2}, \ dots \}}$${\ displaystyle G}$${\ displaystyle z \ in G}$

Now let's look at the sequence at that point . Since the sequence is locally uniformly bounded, it follows from Bolzano-Weierstrasse's theorem that a subsequence exists such that it converges. We call this sequence . ${\ displaystyle (f_ {n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle p_ {1}}$${\ displaystyle f_ {n_ {k}}}$${\ displaystyle f_ {n_ {k}} (p_ {1})}$${\ displaystyle (f_ {1, j}) _ {j \ in \ mathbb {N}}}$

Now you can look at this sequence of functions in the point . With the same argument as above one obtains that there is a subsequence convergent at the point . ${\ displaystyle p_ {2}}$${\ displaystyle p_ {2}}$${\ displaystyle (f_ {2, j}) _ {j \ in \ mathbb {N}}}$

This is how one defines the function sequences inductively . ${\ displaystyle (f_ {i, j}) _ {j \ in \ mathbb {N}}}$

Now consider the diagonal sequence . This converges for all according to Cantor 's diagonal sequence method and is therefore also compactly convergent in the area according to the lemma . ${\ displaystyle (f_ {n, n}) _ {n \ in \ mathbb {N}}}$${\ displaystyle p_ {i} \ in P}$${\ displaystyle G}$

## Individual evidence

1. P. Montel, Sur les familles normales de fonctions analytiques, Annales de l'Ecole Normale Superieure (3), Volume 33, pp. 223-302, 1916.
2. ^ L. Zalcman, Normal families: New perspectives , Bulletin of the American Mathematical Society, Volume 35, pp. 215-230, 1998.