A Method to Prove Approximability by Hahn-Banach & Riesz Representation Theorem

张恭庆，林源渠——泛函分析讲义（上） P153

In this article we give a method which is useful to prove some class of functions can be approximated by other functions by using the famous Hahn-Banach Theorem and Riesz Representation Theorem. In this way, we prove the Runge’s Theorem, which asserts any analytic function can be uniformly approximated by rational functions in some compact region.

Main Theorems and Method

All tools we need are a corollary of Hahn-Banach Theorem , in fact, and the Riesz’s Representation Theorem. Before state our method, we need review these consequences firstly.

Hahn-Banach Theorem

Here we give the usual version of Hahn-Banach theorem in normed space.

Hahn-Banach Theorem (in Normed space)
Given a normed space and its linear subspace , is a bounded linear functional on , then there is a bounded linear functional s.t.

1. (Extension) ;
2. (Norm preserving) .

This theorem tells us there are enough many functionals in to distinguish different elements in , that is, the following corollary.

Corollary 1 If is a normed space, then , s.t.

Inversely, to prove , all we need to do is to show for each . In fact, we can go further and have next corollary in which we are able to not only locate the single point by (), but also distinguish this point from a subspace.

Corollary 2 Given a normed space and is a linear subspace of . If and , then s.t.

1. ;
2. ;
3. .

In short, this normalized functional’s null space includes but not . Such a conclusion sounds like Urysohn Lemma, an important lemma in topology, which says that if the topology space is , then for any closed sets (), there is a continuous function s.t.

Finally, we give the result we really need.

Corollary 3 Given a normed space and a subset in , then we have

if and only if for all satisfying for any , also holds.

The basic idea of this corollary is the functionals on whose null space include can serve as a kind of filter, which can help us to find the limit points of linear space spanned by , and this is exactly what we expect in approximation theory.

Riesz Representation Theorem

What we need here is the Riesz’s representation theorem in a norm space instead of Hilbert space, it helps us to convert the result with general functional to a manageable integration form.

Riesz’s Representation Theorem (in Continous Function Space)
If is a compact Hausdorff space, then (the star indicates it’s the dual space of the normed space ), there is a unique complex Baire measure, that is, a completely addable set-function s.t. and we have

Three Steps of Prove

Step 1: Set to be a large enough space (usually the continuous function space), regard the approximate function space as , spaces of functions to be approximated as . In general, we have .

Step 2: Use the Corollary 3 before, convert the problem of prove

into the problem of showing () for each satisfies .

Step 3: With the help of Riesz’s theorem, prove the integration equation

An example: Runge Theorem

In this section, we use the method given in previous section to prove Runge Theorem:

Runge Theorem
Let be a compact subset in complex plane, denote the extended complex plane as . Assume is a subset of , which insects with every connected component of . If is an analytic function in a neighborhood of , then there is a sequence of rational function, whose poles are located in , converging to in uniformly.

Proof :
Step 1: Let to be the continuous space in , to be the closure of rational functions whose poles are all in , and to be the space of functions satisfying conditions in the theorem.

Step 2: Use the corollary, in order to prove , we only need to show for any , we have for every s.t. .

Step 3: Riesz’s theorem tells us it’s sufficient to prove

where is consisted with all completely addable complex measure.

To have the equation in step 3, we need another lemma:
Lemma , if we denote

then

1. where ;
2. is analytic in ;
3. .

First, we use Cauchy theorem since is analytic in and have

for some closed curve in where is the analytic region of , then

now the lemma enable us to use Fubini theorem to exchange the order of integration to have

If we can show that for each , then the proof is completed. Hence the problem is deduced to discuss on .

The conditions in theorem ensure

where is the connected component of including . Note

Now we show on by proving in every .

Because , is analytic in , i.e. every . Hence we can take the derivative through the integration and have

where for some . On the other hand, we have

hence

If , then the fact of each order of derivative of at is 0 indicates in . If , then we expand at infinity and have

In the end we conclude that , thus
and this completes our proof.