Suppose $f :[a,b]\to \mathbb{R}^n$ is given by $f(t)=\langle f_1(t), \ldots, f_n(t)\rangle$, where each of the $n$ component functions is integrable over $[a,b]$. I think the following inequality
$$\left|\int _a^b f(t) \, dt\right|\leq\int_a^b\left|f(t)\right| \, dt$$
still hold but I don't know how to prove it. Any idea?
Thanks
Best Answer
The inequality you want is a generalization of the triangle inequality. You can either use the triangle inequality at the level of the Riemman sums: $$ \left|\int_a^bf(t)dt\right|=\left|\lim_n \sum_{k=1}^n f(t_{n,k})(t_{n,k}-t_{n,k-1})\right|=\lim_n \left|\sum_{k=1}^n f(t_{n,k})(t_{n,k}-t_{n,k-1})\right|\\ \leq\limsup_n \sum_{k=1}^n |f(t_{n,k})|(t_{n,k}-t_{n,k-1})=\int_a^b|f(t)|dt $$ (where the last equality is justified by the fact that $|f(t)|$ is integrable, being a composition of a continuous function and integrable functions).
Or you can calculate $$ \left|\int_a^bf(t)dt\right|^2=\left\langle\int_a^bf(t)dt,\int_a^bf(t)dt\right\rangle=\int_a^b\int_a^b\langle f(t),f(s)\rangle\,dtds\\ \leq\int_a^b\int_a^b| f(t)|\,|f(s)|\,dtds=\left(\int_a^b|f(t)|dt\right)^2 $$ (where you would still need to justify that the integral of the inner product is the inner product of the integrals, again by the linearity and continuity of the inner product).