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

Example 10.23

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

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

(10.7)

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

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