A \emph{linear functional} on a real vector space $X$ is a linear map $f: X\to\mathbb{R}$. If $M\subseteq X$ is a subspace and $f: M\to\mathbb{R}$ is linear, an \emph{extension} of $f$ to $X$ is a linear functional $F: X\to\mathbb{R}$ such that $F\vert_M=f$.