set is a cone, but it is not necessarily closed.

Again with $p$ a point in $C$, the *solid tangent cone* of $C$ at $p$ is the set of vectors $v$ in $C$ such that $\theta(v){\geqslant}0$ for every functional $\theta$ in $V^*$ that is nonnegative throughtout the translate $C-p$. Since the solid cone is an intersection of sets of the form $\theta^{-1}([0,+\infty))$, which are all closed, the solid tangent cone is a closed cone. It is well known that, conversely, as a consequence of the Hahn-Banach Separation Theorem, the topological closure of the tangent cone is the solid tangent cone.

An intermediate set is the *sequential closure of the tangent cone* of $C$ at $p$, defined to be the set of vectors $v$ in the solid tangent cone at $p$ that can be approximated by sequences of points $\{v_i\}_{i=1}^\infty$ in the tangent cone at $p$, $v_i\to v$ as $i\to \infty$. Again, the sequential closure of the tangent cone is not necessarily topologically closed, and its closure is the solid tangent cone.

We then have

\[thm:characterization\] Let $V$ be a locally-convex topological vector space, and let $C$ be a closed and convex subset of $V$. Given a point $p$ in $C$ 