Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 2 / 3F (Analysis II R1)

The L1 norm properties of absolute homogeneity and the triangle inequality follow from the corresponding properties of the absolute value and the integral. If $\Vert f{\Vert }_1=0$, the nonnegative continuous function $|f|$ has zero integral. Were $|f(x_0)|>0$, continuity would make it bounded below by a positive number on a nontrivial interval, contradicting the zero integral. Thus $f=0$, proving positive definiteness.

Put $M=\Vert f{\Vert }_{\infty }$ and choose the continuous cutoff functionThen $g_n(0)=g_n(1)=0$, $\Vert g_n{\Vert }_{\infty }\le M$, and $f-g_n$ is supported in the two endpoint intervals of total length $2/n$. Therefore

Now suppose ${\int }_0^{1}fg=0$ for every $g\in \mathcal{S}$. Apply this to the above $g_n$. By the uniform norm bound,Hence ${\int }_0^1{f}^2=0$, and continuity again gives $f=0$.