$ cl(A+B)\subset cl(A)+cl(B) $

Question

Let $X$ be a separable Banach space and $A$, $B$ are convex subsets of $X$.

Show that : $$ cl(A+B)\subset cl(A)+cl(B) $$

An idea please.

Answer 1

The result seems false in general.

For any two sets $A,B \subseteq X$ we have $\overline{A} + \overline{B} \subseteq \overline{A+B}$.

In general it is known that the sum of two convex closed sets need not be closed (e.g. there are closed subspaces of a separable Hilbert space whose sum is not closed) so let $A, B \subseteq X$ be two closed convex sets such that $A+B$ is not closed.

If, $\overline{A+B} \subseteq \overline{A} + \overline{B}$, then in fact we have $$\overline{A+B} = \overline{A} + \overline{B} = A+B$$ which would imply that $A+B$ is closed. This is a contradiction.