Let $(C[-1,1],\|.\|_2)$ prove this space isn't a Banach space.

Question

Let $(C[-1,1],\|.\|_2)$ prove this space isn't a Banach space.

My attempt: Let $\{f_n\}$ such that

$f_n(x)= \left\{ \begin{array}{lcc} -1 & if & x \in [-1,\frac{-1}{n}) \\ nx & if & x \in [\frac{-1}{n},\frac{1}{n}) \\ 1 & if & x \in [\frac{1}{n},1] \end{array} \right.$

Note each $f_n$ is a continuous function in $[-1,1]$ then $\{f_n\}\subset C[-1,1]$ Moreover is easy show that $\{f_n\}$ is a Cauchy sequence in $C[-1,1]$.

I'm stuck trying to show that $\{f_n\}$ converges to $f$ but $f\not\in C[-1,1]$.

Answer 1

HINT

Prove that $\|f_n-f\|_2 \to 0$ where $f(x)=-1$ if $x <0$ and $f(x)=1,$ if $x \geq 0$

So the limit is discontinuous at zero,thus it does not belong to the space.

Answer 2

If $f_n \to f$ in your norm, it mist certainly converge pointwise (and the pointwise convergence topology is Hausdorff, so there is at most one pointwise limit). And if the only candidate for a limit is not in the more limited subspace the original sequence cannot converge there.