Show that $E=C([0,1],\mathbb{C})$ is not a hilbert space ...

Question

Show that the prehilbertian space $E=C([0,1],\mathbb{C})$ provided with the scalar product $$ (x|y)=\int_0^1x(t)\bar y(t)dt$$is not a hilbert space!

I know that we have to show that $E$ is not complete but I cannot find any cauchy sequence which does not converge in $E$. Thank you in advance !

Answer 1

Hint: Verify that $f_n(x)=1$ for $0 \leq x \leq \frac 1 2 $ $-\frac 1 n$, $n(\frac 1 2 -x)$ for $\frac 1 2 -\frac 1 n \leq x \leq \frac 1 2$ and $f_n(x)=0$ for $x >\frac 1 2$ defines a Cauchy sequence which does not converge to any element of $C[0,1]$.