Riesz representation theorem

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Theorem: Every continuous linear functional $F$ on the space $C([a,b]\cap\mathbb{T})$ can be represented by a Riemann-Stieltjes $\Delta$-integral of the form $$F(f)=\displaystyle\int_a^b f(t) \Delta\alpha(t),$$ where $\alpha \in \mathrm{BV}([a,b]\cap\mathbb{T})$ and moreover $$||F||=V(\alpha,[a,b]\cap\mathbb{T}).$$

Proof:

References[edit]

[1]