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Let $X$ be a real Banach space and $X^*$ be its dual space. Let $C$ be a weak$^*$ closed subset in $X^*$ and $D$ a compact weak$^*$ in $X^*$. I would like to ask whether $C+D$ is closed weak$^*$ in $X^*$.

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More generally, in a topological group, compact times closed is closed. –  GEdgar Oct 16 '12 at 1:37
    
@GEdgar you mean there is more general result (which you stated)? –  Norbert Oct 16 '12 at 1:40
    
@Norbert: See my answer. –  Nate Eldredge Oct 16 '12 at 1:52

1 Answer 1

up vote 2 down vote accepted

Let $\{c_\alpha + d_\alpha\}$ be a net in $C+D$ with $c_\alpha + d_\alpha \to x$ (in the weak-* topology). Passing to a subnet, we may assume $d_\alpha \to d$. Then $c_\alpha \to x - d$, so since $C$ is closed, $x-d \in C$, which means $x = (x-d) + d \in C+D$. Thus $C+D$ is indeed closed.

Following GEdgar's comment, the same proof shows that in any topological group, the product of a closed set with a compact set is closed.

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Thank you for your solution. –  blindman Oct 16 '12 at 2:20

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