Ok so well Im struggling to find examples for the first two parts and for the last, well I don't think it is open but can't again find an example. Thanks.
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For the last question I'd say yes: let $x=v+w \in V+W$, and $\varepsilon_v,\varepsilon_w>0$ s.t. $B_v=B(v,\varepsilon_v)\subseteq V$, $B_w=B(w,\varepsilon_w)\subseteq W$. Then $B_v+B_w=B(x,\varepsilon_v+\varepsilon_w)\subseteq V+W$. – AndreasT Nov 14 '13 at 10:12
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First question is duplicated but other two are not. – user52045 Nov 14 '13 at 10:17
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@user52045 Yes. I noticed that I will put the answer of the second question in a minute. But I really think I saw it being asked here before – Amr Nov 14 '13 at 10:17
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1@WhizKid Please do not ask multiple questions at once. This makes your question unlikely to be of help for future readers (who are unlikely to have trouble with precisely the same set of questions), and also harder to answer all of them in satisfying detail. – Lord_Farin Nov 14 '13 at 12:00
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Since the second question is not answered in the link I gave, I will put its answer here. If $X$ is closed and bounded then its compact by Heine Borel theorem. Let $x_n+y_n$ be a convergent sequence in $X+Y$. Since $X$ is compact, therefore $\{x_n\}$ has a convergent subsequence $\{x_{n_k}\}$ that converges to $x$. $x$ must lie in $X$ , because $X$ is closed. Since $x_n+y_n$ converges, therefore $x_{n_k}+y_{n_k}$ converges as well. Since $x_{n_k}$ converges and $x_{n_k}+y_{n_k}$ converges, therefore $y_{n_k}$ converges to a limit say $y$. $y$ must lie in $Y$, because $Y$ is closed. Therefore the limit of $x_n+y_n$ which is $x+y$ must be inside $X+Y$, because $x\in X$,$y\in Y$
Amr
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