In the case finite-dimensional, for example, $A\subset \mathbb{R^n}$ convex, we know that $\mathrm{cone}(A)$ the cone enerated by $A$ is defined by 
$$\mathrm{cone}(A):=\left\{ \left.\sum\_{i=1}^{m}\alpha\_{i}x\_{i} \:\right|m\leq n+1,\: x\_{i}\in A, \: \alpha\_{i}\geq 0 \:\:\forall i=1,\ldots,m,\:\: \sum\_{i=1}^{m}\alpha\_{i}=1\right\}.$$

**The Question:** Let $X$ be a ***infinite-dimensional*** vector space and $A\subseteq X$ convex set. ***Is there a definition of $\mathrm{cone}(A)$ the cone generated by $A$?***