TITLE: A criterion for the sum of two closed sets to be closed ?
QUESTION [11 upvotes]: 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 ?

REPLY [2 votes]: In finite dimensional space, your result is true. Take a linear functional with $\text{ker } T = I$. It is not hard to prove, using $I\cap V={\boldsymbol 0}$, that $T(V)$ is closed by seeing that $$\text{inf }_{\begin{array}{c}\boldsymbol x \in V \\  \|\boldsymbol x\| = 1\end{array}} \|T(\boldsymbol x)\| > 0,$$ with the minimum achieved. Thus, by Lemma H1 of the book Geometric Functional Analysis and its Applications, $I+V$ is closed.