Let A,B be subsets of a normed linear space X.
Show that 1) If A or B is open then A+B is open.
2) If A and B both are compact then A+B is compact..
Any help appreciated..
1)are you sure it's not if A and B is open?
2) Assuming both A and B are both compact, then for any open cover $\displaystyle {O_a}| a \in S $ covering A we can find a finite subset G of S s.t $\displaystyle {O_p} | p\in G $ is also an open cover for A. same goes for B... Is the union of two finite sets finite?
Suppose that A is open. Let $\displaystyle a+b\in A+B$ and $\displaystyle d\in X$ with $\displaystyle \|d-(a+b)\|<\delta$. Then $\displaystyle \|(d-b)-a\|<\delta$, which implies that $\displaystyle d-b\in A$ if $\displaystyle \delta$ is sufficiently small (because A is open). Then $\displaystyle d = (d-b)+b\in A+B$. That shows that A+B is open.