The existence of unbounded closed convex sets with trivial recession cone in normed spaces (Q300663)

Given a nonempty subset \(C\) of a normed space \(X\), its recession cone is the set \(O_X^+(C) := \{v\in X: C + \lambda v\subseteq C, \;\forall \lambda > 0\}\). It is well known that a closed convex set in a finite-dimensional normed space is unbounded if and only if its recession cone \(O_X^+(C)\) consists of the zero vector alone. A question raised naturally is whether this characterization of the unboundedness remains true in any normed space. The author shows that in every infinite-dimensional normed space there exists an unbounded closed convex set \(C\) for which \(O_X^+(C) \neq \{0\}\).