Norm of a Dirac's functional

Question

Let be $\delta:(\mathcal{C}[-1,1], \| \quad \|_{sup})\to(\mathbb{R}, \lvert \quad \rvert)$ defined as $\delta:=2\delta_{-1}-3\delta_0+\delta_1$ with the Dirac functional $\delta_c\in\mathcal{C}[-1,1]'$ for $c=-1,0,1$. Calculate $\|\delta\|$.

We know that $\|\delta\|\le2\|\delta_{-1}\|+3\|-\delta_0\|+\|\delta_1\|\le6$, but how can I calculate the exact value of $\|\delta\|$?

Answer 1

You know $\|\delta\|\le 6$. Can you find some $f$ such that $\|f\|_{\sup}=1$ and $\delta(f)=6$? This should not be hard to find.