I need help with the following exercise, leading to the conclusion that $X := C([-1,1])$ (space of continuous functions $f: [-1,1] \to \mathbb R$) equipped with the $\lVert \cdot \rVert_1$-norm is not complete:
For each $n\in \mathbb N$, define $\delta_n : X \to \mathbb C$ by $$\delta_n(f) := \frac{n}{2} \int_{-1/n}^{1/n}f(x)\,dx.$$ Prove:
1) $\delta_n \in X'$ for each $n\in \mathbb N$ and $\lVert \delta_n \rVert = \frac{n}{2}$ (the last norm is the operator norm).
2) $\delta_\infty : X \to \mathbb C, \delta_\infty(f) := f(0)$ does not define a bounded linear functional on $X$.
3) $\lim_{n \to \infty} \delta_n(f) = f(0) $ for all $ f \in X$.
4) $X$ is not complete.
Attempt:
1) To show that $\delta_n(f)$ is bounded is straight forward. However, how do I compute the operator norm? I showed $$\lVert \delta_n \rVert \leq \frac{n}{2}$$
but how do I get equality?
2) I tried to find a bounded sequence $f_n$ for which $(\delta_\infty(f_n))_n$ is unbounded, but couldn't think of something that worked out.
3) No problems here.
4) How does this follow from the other points?
Hints:
1) See 2.
2) Use a function which graph is a pointed hat having height $n$ and basis $\frac{2}{n}$
4) If $X$ is a Banach space, the uniform boundedness principle applies to the sequence $\delta_n$