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Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$

a). Show that $\|\cdot\|$ is a norm in $C[a,b]$

b). Show that $C[a,b]$ with this norm is not a Banach space.

Well the 1st question is Kind of easy. Non-negativity, Homogenity, and Triangle inequality is easily proven. At the 2nd question I have a Problem.

I know the $$C[0,1]$$ space with Norm $$\|f\| = \int_0^1|f(x)|d(x)$$ is not complete that means Not Banach-Space. But how do I prove the generality?

Does anyone has a suggestion?

Thanks

Martin Sleziak
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suugii19
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    You find explicitly a sequence of continuous functions that is Cauchy with respect to this norm, but which does not converge in the same norm. – John B Apr 12 '16 at 15:45
  • You mean i have to find an example that contradicts the Norm – suugii19 Apr 12 '16 at 15:46
  • He means that you should find a sequence that looks like it's supposed to converge, but where there is no continuous limit function. – Arthur Apr 12 '16 at 15:48
  • Somewhat similar question: http://math.stackexchange.com/questions/402841/c0-1-is-not-complete-space-with-respect-to-norm-lvert-f-rvert-1-int?lq=1 – Martin Sleziak Apr 12 '16 at 15:48
  • now i get it thanks Arthur – suugii19 Apr 12 '16 at 15:49

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