Exercise 3.7：Let $E$ be a banach space and let $A \subset E$ is a closed subset in the weak topology $\sigma(E,E^{*})$. Let $B\subset E$ is a compact subset in the weak topology $\sigma(E,E^{*})$.
Blockquote 1.Prove that $A+B$ is closed in $\sigma(E,E^{*})$.
Following is the answer of Brezis:
- Let $x \not\in A+B$. We shall construct a neighborhood $W$ of $0$ for $\sigma(E,E^{*})$ such that $$ (x+W)\cap (A+B)=\emptyset. $$ For every $y\in B$ there exists a convex neighborhood $V(y)$ of $0$ such that $$ (x+V(y))\cap(A+y)=\emptyset. $$ (Since $A+y$ is closed and $A\not\in A+y$). Cleary, $$ B\subset \cup_{y\in B}(y-\frac{1}{2}V(y)). $$
Until here, I don't konw why $B\subset \cup_{y\in B}(y-\frac{1}{2}V(y)).$ How to prove this?
