Let $V$ and $I$ be two closed subsets of a Banach space $A$.


The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.


I would like to know whether $I+V$ is closed. I've seen that there is a criterion of Dieudonné which I can't use here because I know that neither $V$ nor $I$ is locally compact.


So my question is : are there any other criteria that I could try to use ? 



A criterion for the sum of two closed sets to be closed ?
When $V$ is also a subspace, the standard equivalence to the sum being closed is that the unit spheres of $V$ and $I$ are a positive distance apart. I bet this is true when $V$ is just a convex cone but don't have time right now to think about it (the given condition is clearly sufficient for closedness of the sum; necessity is the direction that requires thought).


EDIT: It will come to no surprise to those who know me that I lose my bet. Let $A$ be $\ell\_p$, $1\le p < \infty$ ($c\_0$ is also OK with suitable notational changes) and $x^\*$ the linear functional $\sum 2^{-n} e\_n$, where $e\_n$ is the unit vector basis for $A^\* = \ell\_q$, $1/p + 1/q = 1$. $I$ is the kernel of $x^\*$ and $V$ are the non negative vectors in $A$. In Pietro's notation, $\pi(V)$ is closed because it is a cone which contains its base point in a one dimensional space and hence, as Pietro remarked, $V+I$ is closed in $A$. For $n>1$ let $x\_n=e\_n-2^{-n+1}e\_1 \in I$ and observe that $e\_n-x\_n\to 0$.


