where $\mathcal{P}=\{{x}_{0}\text{,}\dots \text{,}{x}_{n}\}$ is a partition of $[a\text{,}b]$.

Example 10.23

Given fixed $f\in C(a\text{,}b)$, the function

$F(u)={\int}_{a}^{b}f(x)\mathit{du}(x)\text{,}u\in \mathit{BV}(a\text{,}b)$ (10.7)

(10.7)
is a continuous linear functional on $\mathit{BV}(a\text{,}b)$. The linearity follows from Theorem 10.4(b), and the continuity ...