Take the 2-minute tour ×  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 ?  share|improve this question Maybe I should add that by
"criterion" I mean a sufficient condition. –  Fabien Besnard Feb
14 '12 at 15:51  3 Answers 3  up vote 4 down vote accepted  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 surpr