Let $V$ be an infinite-dimensional real vector space, and $C\subseteq V$ a non-empty convex cone. Recall that $C$ is said to be algebraically closed if the following condition holds: for all $x,y\in V$, $[x,y)\subseteq C\implies y\in C$.
Suppose now that $C$ is algebraically closed, and that $W$ is a linear subspace of $V$ such that $W\cap C=0$. Does it follow that $W+C$ is algebraically closed?
(This would follow if $V$ were finite-dimensional.)