0

I have to prove that the space of continuous functions on a (I suppose closed) interval $I$ (let's say $[0,1]$) with distance $d(f,g)$ $=$ $\int _I$ $|f-g|$ is not complete. The Cauchy function to be used is $f_n$ = $n$ in [$0$, $e$ $^-$ $^n$] and $\log(1/x)$ in [$e$ $^-$ $^n$, $1$]

Why is it not complete though? Thanks!

Martin Sleziak
  • 53,687
Rock
  • 19
  • What does it mean to not be complete? – user217285 Sep 01 '16 at 08:26
  • So is your sequence a Cauchy sequence? What is it's limit in $L^1$? –  Sep 01 '16 at 08:26
  • Not complete means it doesn't converge in the space... It is Cauchy and it converges to $log(1/x)$ which actually is not bounded in $0$. So the unboundedness is sufficient to say it's not complete? – Rock Sep 01 '16 at 08:38
  • Sort of, because $\log (1/x)$ is not in $C[0,1]$. –  Sep 01 '16 at 08:50
  • Because I cannot find $x$ close to 0 s.t. $log(1/x)$ is... any number I could arbiitrarily assign to 0? (I realise this is a pretty bad question to ask) Thanks so much for your help! – Rock Sep 01 '16 at 08:59
  • See also: http://math.stackexchange.com/questions/410745/ – Watson Sep 01 '16 at 09:15
  • The hard part is showing that it cannot converge in the integral sense to some continuous function. Let $f \in C[0,1]$ be given. For every $\epsilon > 0$ and every positive integer $N$, trying showing that there exists $n > N$ such that $d(f,f_n) > \epsilon$. – Disintegrating By Parts Sep 01 '16 at 17:59

0 Answers0