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