How to get this norm of the functional

Question

Let $\mathcal{C}$ denote the space of $L^\infty(\mathbb{R})$ consisting of continuous bounded functions on $\mathbb{R}$. Define the linear function $\ell_0$ on $\mathcal{C}$ by $$\ell_0(f) = f(0), f\in \mathcal{C}$$ Then $$|\ell_0(f)| \leq \Vert f\Vert_{L^\infty}$$ How to get this result?

Answer 1

It's totally straightforward: $$|\ell_0(f)|=|f(0)|\leq\|f\|_{L^\infty}.$$ The latter bound follows since $f$ is continuous and bounded, so $$|f(x)|\leq \sup_{y\in\mathbb{R}}|f(y)|=\|f\|_{L^\infty}<\infty,$$ for all $x\in\mathbb{R}.$ The equality $$\sup_{y\in\mathbb{R}}|f(y)|=\|f\|_{L^\infty}=\inf_{C\geq 0}\{|f(y)|\leq C\text{ for almost every } y\}$$ is true because $f$ is continuous. If you don't know this result, see Infinity norm of continuous function.