Let $X=(X,|\cdot|)$ be a Banach space and let $A,B \subset X$. If $A$ is a bounded set and $B$ is a compact set, then $$A+B=\{a+b \in X \; ; \; a \in A \: \text{e}\: b \in B\}$$ is a totally bounded set?
Is this true or not? If not, is there any additional hypothesis that makes it true? I couldn't prove it or think of a counterexample.
This is not true in general. Take an infinite dimensional Banach space $X$, let $A$ be the closed unit sphere and let $B=\{0\}$. Then $A$ is bounded and $B$ is compact. But $A+B=A$, which is not totally bounded (since, if it was and since $A$ is complete, $A$ would be compact).