Here's a simple answer: Since the L_2-norm is weaker than the sup-norm and C[a,b] is dense in L_2[a,b], the polynomials are dense there as well.

The space of polynomials is dense in L^p([a,b],\mu), 1\leq p< \infty,

where \mu is measure.

The question posed by Abdelmalek can be regarded in the more general context of metric spaces, and the answers by Werner and Mateljevic as examples in the following set-up:

1) Consider both spaces C[a,b] and P[a,b] as metric spaces with the sup-metric, so P[a,b] is dense and continuously embedded into C[a,b];

2) Let rho be any metric on C[a,b] which is weaker than the sup-metric;

3) Let X be the "completion" of C[a,b] under the metric rho.

Then, P[a,b] is dense in X.

As professor David G. Costa said

''the question posed by Abdelmalek can be regarded in the more general context of metric spaces''.

We add that the subject is very interesting and related to the other subjects. Here we shortly discuss Mergelyan's theorem.

Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Nikitovich Mergelyan in 1951. It states the following:

Let K be a compact subset of the complex plane C such that C\K is connected. Then, every continuous function f : K\to C, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K.

Mergelyan's theorem is the ultimate development and generalization of the Weierstrass approximation theorem and Runge's theorem. It gives the complete solution of the classical problem of approximation by polynomials.

See W. Rudin, Real and Complex Analysis, McGraw–Hill Book Co., New York, (1987), ISBN 0-07-054234-1.

First of all, I would like to thank D. Werner, M. Mateljevic, and D. Costa for the valuable insights. I will summarize what you have said and please do tell me if my understanding is flawed.

Let me first define the following spaces:

- C(K) is the set of all complex valued continuous functions in the compact set K equipped with the supremum norm, which constitutes a Banach space.

- C_2(K) is the set of all complex valued continuous functions in the compact set K equipped with the Euclidean norm, which is not complete.

- L_2(K) is the space of all measureable complex-valued functions whose square is integrable equipped with the Euclidean norm, which is complete.

- P[z] is the space of all complex-valued Laurent polynomials.

The space P[z] along with the supremum norm is dense in C(K) and along with the Euclidean norm is dense in L_2(K).

Is my understanding correct?

Also, I would appreciate any references that may help me comprehend the subject.

Kind regards,

Salem.

Mergelyan's theorem is a famous result from complex analysis proved by the Armenian mathematician Sergei Nikitovich Mergelyan in 1951. It states the following:

Let K be a compact subset of the complex plane C such that C\K is connected. Then, every continuous function f : K\to C, such that the restriction f to int(K) is holomorphic, can be approximated uniformly on K with polynomials. Here, int(K) denotes the interior of K. 

Here  a  complex polynomial P in one indeterminate z over a ring C of complex number is defined as a formal expression of the form

$P=P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z^1 + a_0$

where n is a natural number, the coefficients a_0, . . ., a_n are elements of $C$.

If $K= {1\leq |z|\leq 2}$ is annulus ( in this example,  C\K is not  connected)  then function $J(z)= 1/z$ can not be uniformly approximated by complex polynomials.

See also Runge's theorem.

We need to define  the space of all complex-valued Laurent polynomials.

Is it  related to   the set of 

rational functions?

Dear S. Abdelmalek,

Set  

(I)   ''The space P[z] along with the supremum norm is dense in C(K) and along with the Euclidean norm is dense in L_2(K).''

At first glance it seems that (I) is related to   Runge's approximation theorem.

Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:

Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every bounded connected component of C\K then there exists a sequence (r_n)_{n\in\N} of rational functions which converges uniformly to f on K and such that all the poles of the functions (r_n)_{n\in\N} are in A.

Note that not every complex number in A needs to be a pole of every rational function of the sequence (r_n)_{n\in\N}. We merely know that for all members of (r_n)_{n\in\N} that do have poles, those poles lie in A.

One aspect that makes this theorem so powerful is that one can choose the set A arbitrarily. In other words, one can choose any complex numbers from the bounded connected components of C\K and the theorem guarantees the existence of a sequence of rational functions with poles only amongst those chosen numbers.

A possible interesting answer, but for X=(L^1)(A, miu) is valid for arbitrary closed unbounded subsets of R^n, miu being a positive regular Borel Moment-determinate  measure on A (miu is uniquely determinate by its moments, or, equivalently, by its values on polynomials). Moreover, one can prove that the nonnegative polynomials on such a subset A are dense in the positive cone of X. Precisely, for any nonnegative continuous vanishing at infinity function "phi" on A, there exists a sequence (p_m) of polynomials, p_m > =( "phi" ) for all m, p_m converging to "phi" in the norm on X. For the proof see "Approximation and Markov moment problem on concrete spaces" by Octav Olteanu, Rendiconti del Circolo Matematico di Palermo, Springer Verlag Italia, (2014), DOI 10.1007/s12215-014-0149-7

Dear S. Abdelmalek,

The Hardy space H_2 consists of the functions f whose mean square value on the circle of radius r remains bounded as r tends 1 from below.

More generally, the Hardy space H_p for $0 < p < \infty$ is the class of holomorphic functions f on the open unit disk satisfying that the integral means $M_p(f,r)$

on the circle of radius r remains bounded as r tends 1 from below.

The complex polynomials are dense in the Hardy space H_p for $0 < p < \infty$.

It is possible to define Hardy spaces on other domains than the disc, and in many applications Hardy spaces on a complex half-plane (usually the right half-plane or upper half-plane) or on the annulus are used.

Let $K =K_R={z: 1/R< |z|

Dear Colleagues,

There are hosts of approximation theorems by polynomials in the contemporary literature. There a lot books on the subject as well (google them!). Stone-Weierstrass theorem is just the beginning. Great deal of the approximation theory has been devoted to the subject of polynomial approximation.

We can, of course, append more and more theorems here, but the question asked by S. Abdelmalek here (despite its simplicity of formulation) is too general to provide a concise answer.

 Dear all,

I must say that it is nice when a question like the one asked by Abdelmalek invites the so many good answers we read. At the same time, Sznajder has a point by saying the question is too broad to have a short concise answer. But of course we all enjoyed the exchange of ideas.

Dear all,

Thanks again for all your insights. Attached is a pdf file containing the detailed question. I hope this clarifies what I want to say.

Dear S. Abdelmalek,

Concerning your file Question3.pdf  we can say:

The function $\underline{p}' (z)$ belongs to $L_2(K)$, where $K ={z: 1/R< |z|