TITLE: Basis of $C[a,b]$
QUESTION [6 upvotes]: 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 ?

REPLY [9 votes]: 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.