Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
2026-05-04 09:59:55.1777888795
Continuous functions. Second norm
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Clearly $$ \int_a^b f(x)\,dx=\lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^n f(t_{n,i}), $$ where $t_{n,i}=a+i(b-a)/n$, and $$ \Big\|\sum_{i=1}^n f(t_{n,i})\Big\|_2\le \sum_{i=1}^n \|f(t_{n,i})\|_2, $$ due to the triangular inequality. But $$ \lim_{n\to\infty}\frac{b-a}{n}\sum_{i=1}^n \|f(t_{n,i})\|_2=\int_a^b\|f(x)\|_2\,dx, $$ etc...