Let $V$ be a real [vector space](/page/Vector%20Space), let $C \subset V$ be a convex set, and let $I$ be a nonempty index set. For each $i \in I$, let $f_i:C\to\mathbb{R}$ be a [convex function](/page/Convex%20Function). Define the extended-valued function $f:C\to\mathbb{R}\cup\{+\infty\}$ by