Basis of $C[a,b]$

Question

The space $C[a,b]$ , space of all real valued continuous functions on $[a,b]$ is an infinite dimensional vector space over the field $\Bbb R$. As every vector space over a field has a basis so definitely $C[a,b]$ has a basis. I want to know a basis of $C[a,b]$. How I can find a basis of it ?

Answer 1

If you're looking for a Schauder basis, meaning a basis from which every vector may be written as an infinite sum, then by the Stone-Weierstrass theorem, the polynomials are dense in $C[a,b]$ in the supremum norm. That means every continuous function can be written as an infinite sum of polynomials. For example Bernstein polynomials are a convenient spanning set. But if you don't like polynomials, many other families of functions also serve. The Stone-Weierstrass theorem applies to any family of functions that separates points. Trig functions, exponentials, absolute values, whatever you want. And there is a process for turning these spanning sets into bases.

But if you want an actual algebraic basis in the sense of linear algebra, a Hamel basis, a basis such that every vector is a finite linear combination, well such things exist by the axiom of choice, but this is nonconstructive. No explicit formula for such bases may be given in general, and it is even consistent with ZF set theory that they not exist.

Answer 2

You mention that every vector space has a basis, which is a fact about Hamel bases, but your question is tagged with shauder-basis. Which one are you asking about? I'll just answer both and highlight the differences.

A Hamel basis is what we usually mean when we just say "basis". I.e. every element can be uniquely written as a finite linear combination of basis elements. Necessarily, any Hamel basis of $C[a,b]$ will be uncountable. See What is a basis for the vector space of continuous functions? for why there is essentially no way to write down a Hamel basis of $C[a,b]$ without invoking the Axiom of Choice.

A Schauder basis is a sequence $(f_n)$ such that for any element $f$ of the space, there is a unique sequence of constants $(\alpha_n)$ such that $f = \lim_n \sum_{i=1}^n \alpha_n f_n$, where the limit is taken with respect to the norm of the space (in this case, the sup norm on $C[a,b]$). The most common Schauder basis for $C[0,1]$ is a certain type of wavelet basis, called a Faber–Schauder system. Describing such a system is a task that I don't attempt here, but I encourage reading the linked wiki.