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

Put $I={\int }_0^{1}f(t)dt$. If $I\ne 0$, take the inner product with the unit vector $u=I/\Vert I{\Vert }_2$. Thenby the Cauchy-Schwarz inequality; the case $I=0$ is immediate. This is the triangle inequality for a vector-valued integral.

Equality in the Euclidean argument requires $u\cdot f(t)=\Vert f(t){\Vert }_2$ for every $t$: the nonnegative continuous difference has integral zero. Equality in Cauchy-Schwarz inequality then says $f(t)=\lambda (t)u$ with $\lambda (t)\ge 0$. Conversely, if $f(t)=\lambda (t)v$ for one fixed $v\in {\mathbb{R}}^m$ and a continuous $\lambda \ge 0$, then for every norm,Thus these nonnegative-ray functions, including $f\equiv 0$, are exactly the required functions.