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I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm.

I guess I need to show that every Cauchy sequence in the space, does not converge under the given norm. But I am a bit lost on how, not asking for full solutions here. Just some tips on how to get started. Maybe some general tips on how tackle such problems?

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    Show that the indicator function of the interval $[\tfrac{1}{2}, 1]$ is an element of the $L^2$ closure. Try the first sequence of continuous functions that springs to mind.2012-11-07
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    Well, it suffices to find a Cauchy sequence that does not converge. Note that you're talking about continuous functions here, so a Cauchy sequence that has a non-continuous pointwise limit will do the trick, since although it converges to a limit in a bigger class of functions, that limit isn't in the space you're considering.2012-11-07
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    It's not entirely correct that you need to show that **every** Cauchy sequence does not converge in that space, but merely, **some** don't. There are Cauchy sequences that do converge, you just need to find one that doesn't.2012-11-10

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You need to find a $\|\cdot\|_2$-Cauchy sequence which doesn't converge in $C([0,1])$. You can think of approximating a jump function (with jump in the interior of $[0,1]$) by piecewise affine functions.

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    I am thinking something along the lines of a trigonometric sequence. What about $x \Rightarrow \sin(1/x)$?2012-11-07