TITLE: What does $C[0,1]$ mean?
QUESTION [6 upvotes]: In the context of real analysis, I have found this question: 
For each $$f \in C[0,1] $$ there is a series of even polynomials , which converge uniformly on $[0,1]$ to f. 

What is $C[0,1]$ ? Is it the space of functions which are continuous for $0\le x \le 1 $ ?

REPLY [11 votes]: Yes it is.  It is the space of all continuous functions from $[0,1]$ to $\mathbb{R}$.  It has some mathematical structures under some specified operations. For example, $C[0,1]$ is a vector space over the field of reals.
In the space $C[0,1]$, points are just continuous functions.  You can define operation on them like $(f+g)(x) =f(x)+g(x) = (f+g)(x)$  and multiplication like $(fg)(x)=f(x)g(x)=(fg)(x)$. These are called pointwise addition and pointwise multiplication.

REPLY [3 votes]: $C[0,1]$ is the set of continuous functions on the closed interval $[0,1]$.