Is $C([0,1])$ a compact space?

Question

Is $C([0,1])$ (I guesss with the max-norm) a compact space?

I have to know that because I want to apply Arzela Ascoli.

Answer 1

Consider $f_n(x) = \begin{cases} 0 & x \in [0,\frac{1}{2}-\frac{1}{n}) \\ 1+n(x-\frac{1}{2}) & x \in [\frac{1}{2}-\frac{1}{n}, \frac{1}{2}) \\ 1 & x\in [\frac{1}{2},1] \end{cases}$.

(Since you were considering Arzela Ascoli.)

Answer 2

No, of course not. No (nontrivial) normed space is compact. The sets $\{v\mid\|v\|<n\}$ form an open cover with no finite subcover.

Answer 3

No it is not, but you don't need $\rm C([0,1])$ to be compact to apply Ascoli-Arzela, but that $[0,1]$ is compact !