To be used in tutorial

The following is a standard result for functions of bounded variation. If $g$ is continuous then we can still show this without resorting to any known result for functions of bdd variation:

Problem. Let $g\in C[a,b]$ and let $\Lambda:C[a,b]\to \mathbb R$ be a linear function given by $$\Lambda(f)=\int_a^b f(x)g(x)\,dx.$$ Show that $$\|\Lambda\|:=\sup\left\{\left|\int_a^b f(x)g(x)\,dx\right|: f\in C[a,b], \sup_{x\in [a,b]}|f(x)|\leq 1\right\}=\int_a^b |g(x)|\,dx.$$