Let $\chi:[-1,+1]\to \mathbb{R}$ be the characteristic function of interval $[0,1]$, i.e. $$ \chi(x)=\left\{\begin{array}{ccc} 1 &\mbox{if} & x\in [0,1]\\ 0 &\mbox{if} & x\notin [0,1] \end{array}\right. $$ I would like to prove that there is a continuous linear functional $\varphi: E=L^{\infty}([-1,+1])\to \mathbb{R}$ such that
- $\|\varphi\|_{E^\ast}=2 $,
- $\varphi(\chi)=1$,
- $\varphi(f)=0$ for all $f$ continuous.
My attempt. Let $G\subset L^{\infty}([-1,+1])$ of sectionally continuous functions in $L^{\infty}([-1,+1])$. I tried to define $\varphi$ in G by putting $\varphi(g)=\varphi_+(g)-\varphi_-(g)$ for $$ \varphi_+(g)=\lim_{t\to 0^+}g(t)\quad \mbox{ and } \quad \varphi_-(g)=\lim_{t\to 0^-}g(t). $$ Then I try to use the Hahn-Banach theorem to extend the functional $ \varphi$ from $G$ to $L^\infty([-1,1])$. But I have no control over the norm of $\varphi $.
