# Topological space of continuous function is not compact

I'm struggling with this question:

Let $C[0,1]$ be set of continuous function of $[0,1]$. Define metric $d(f,g)=\int^1_0|f(x)-g(x)|dx$. Show that $C[0,1]$, with topology $\tau$ induced by $d$, is not a compact space.

Thanks for your help in advance.

• Do you mean compact or complete? – Hamou Aug 18 '14 at 2:50
• I mean compact. – A. Chu Aug 18 '14 at 3:04

## 1 Answer

This is a (normed) vector space ($d$ result for a norm) , hence not bounded, so not compact.

• How about proving from the definition? (Open cover doesn't has finite subcover) – A. Chu Aug 18 '14 at 3:14
• Use the cover $\{B(f,1)\}_{f\in C[0,1] }$, there is no finite sub-cover. Where $B(f,1)$ is the open ball of center $f$ and radius $1$. – Hamou Aug 18 '14 at 3:17
• In general, every (non-zero) normed vector space $(E,\|.\|)$ is not compact. – Hamou Aug 18 '14 at 3:28
• @Hamou Is it ture? How can we prove that? – Bumblebee Aug 18 '14 at 8:06