Halfspaces are Convex Sets
# Statement 1
Let $V$ be an Inner Product Space over $\mathbb{R}$, let $a \in V$, $b \in \mathbb{R}$, and let $H$ be the Closed Halfspace $$H = {x : \langle x, a \rangle \geq b}.$$ Then $H$ is a Convex Set.
# Proof
TODO This should follow from the fact that Inner Products are Linear in their first argument
# Statement 2
Let $V$ be an Inner Product Space over $\mathbb{R}$, let $a \in V$, $b \in \mathbb{R}$, and let $H$ be the Open Halfspace $$H = {x : \langle x, a \rangle > b}.$$ Then $H$ is a Convex Set.
# Proof
TODO follows from statement 1 and Interior of Convex Set is Convex.