Convex Subset of the Lebesgue Space of Square-Integrable Vector-Valued Functions

Let $T>0$ and $d$ be as in the definition of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$, whose vector operations are those of that definition.

A subset $C$ of $L^{2}([0,T];\mathbb{R}^{d})$ is convex if $su+(1-s)v\in C$ for all $u,v\in C$ and every real number $s$ with $0\le s\le1$.