Corollary 2

If C is a pointed convex cone, the largest vector subspace contained in C is the set $C \cap (-C)$.

For, if $W = C \cap (-C)$, then W is non-empty and $\lambda W = W$ for all $\lambda \neq 0$, also

which shows that W is a vector subspace. Clearly every vector subspace contained in C is also contained in W.

Obviously, if $f$ is a linear mapping of E in a vector space F, then $f(C)$, the image of a convex cone C in E, is a convex cone in F. Every intersection of convex cones (with vertex 0) in E is a convex cone. For every subset A of E the intersection of convex cones containing A (these exist, E itself is one such cone) is the smallest convex cone that contains A; it is called the convex cone generated by A.