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Let $F \subset \mathcal{C}([0, 1], \mathbb{R})$ be closed, equicontinuous, and pointwise bounded. Let $I : F → \mathbb{R}$ be defined by $I(f) = \int\limits_0^1 f(x)dx$. Show that there is $f_0 \in F$ such that $I(f_0) ≥ I(f)$ for all $f \in F$.

By Ascoli's theorem, $F$ is clearly compact. Do I then need to show that there is a maximal element?

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By Arzela-Ascoli you have that $F$ is compact, since $I$ is a continuous linear functional you have by the Extreme Value theorem that $\displaystyle \sup_{f\in F}\;I(f)$ is achevied for some $f_0\in F$.