Infinite dimensional convex cones

Question

Let $C$ be a convex cone in a topological real vector space $V$. Assume that we have a linear functional $\varphi: V \to \mathbb{R}$ such that $\varphi(x) \geq 0$ for all $x \in C$. Further assume that $x \in C$ and $\varphi(x)=0$ implies $x=0$. If $V$ is finite dimensional with euclidean topology, then the set $\{x \in C:\, \varphi(x)=1\} \subseteq V$ is compact. Is there an easy example where this fails in infinite dimension?

Answer 1

In the space $\ell_1 $ consider the cone $V=\{ x\in\ell_1 : \forall_i x_i \geq 0\}$ and functional $\varphi :\ell_1 \to \mathbb{R} ,$ $\varphi (x) =\sum_{j=1}^{\infty} x_j .$