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I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous.

Any ideas?

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Instead of "converges under $\|\cdot\|_2$ to a limit which isn't continuous," you probably want "does not converge under $\|\cdot\|_2$ to a limit which is continuous." – Jonas Meyer May 11 '12 at 19:19
up vote 5 down vote accepted

Let $$f_n(x) = \left\{ \begin{array}{rl} 0 & \text{if } x \leq 1/2,\\ 1 & \text{if } x \geq 1/2+1/n,\\ n(x-1/2) & \text{if } 1/2\leq x\leq 1/2+1/n. \end{array} \right.$$

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Does this sequence work for showing C[0,1] is not Banach for any $p<\infty$? I can see it works for $p=1$ & $p=2$. – rk101 May 11 '12 at 13:23
Yes, I think so. – Michael Greinecker May 11 '12 at 13:25
Yes it does actually, I have just checked it. – rk101 May 11 '12 at 13:38
A quick question: Why $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ ? – rk101 May 11 '12 at 13:56
I'm not completely ure. I think for simple functions you can use the corresponding reult for finite dimensioal spaces and then take limits. – Michael Greinecker May 11 '12 at 14:23

Another one. Think of a discontinuous (bounded measurable) function. Say: $f(x) = 0$ on $[0,1/2]$ and $f(x) = 1$ on $(1/2,1]$. Write down its Fourier series. The partial sums are continuous. They converge in $L_2$ norm (to $f$) but do not converge to any element of $C[0,1]$.

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Note that $f$ should be not only discontinuous, but not a.e. equal to any continuous function. E.g. $f(1/2) = 1$, $f(x) = 0$ otherwise is discontinuous, but its Fourier series converges uniformly to the continuous function 0. – Nate Eldredge May 11 '12 at 13:10

Think of the function that is $0$ on the interval $\left[0,1-\frac{1}{n}\right]$ and then is $y=n(x-1)+1$ on the remainder of the interval. As $n$ increases the "spike" gets sharper. The limit function is not continuous at $x=1$.

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However, the limit function in the $L^2$ norm is $0$, which is in $C[0,1]$. These aren't the pointwise limits you're looking for... – robjohn May 11 '12 at 15:33

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