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.

  • 1
    $\begingroup$ Do you mean compact or complete? $\endgroup$ – Hamou Aug 18 '14 at 2:50
  • $\begingroup$ I mean compact. $\endgroup$ – A. Chu Aug 18 '14 at 3:04

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

  • $\begingroup$ How about proving from the definition? (Open cover doesn't has finite subcover) $\endgroup$ – A. Chu Aug 18 '14 at 3:14
  • $\begingroup$ 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 $1$. $\endgroup$ – Hamou Aug 18 '14 at 3:17
  • $\begingroup$ In general, every (non-zero) normed vector space $(E,\|.\|)$ is not compact. $\endgroup$ – Hamou Aug 18 '14 at 3:28
  • $\begingroup$ @Hamou Is it ture? How can we prove that? $\endgroup$ – Bumblebee Aug 18 '14 at 8:06

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