I have read that if the subsets $A$ and $B$ of a topological vector space are bounded, i.e. for any neighbourhood $U$ of $0$ there is an $n>0$ such that, for all $|\lambda|\geq n$, $M\subset\lambda U$, then $A+B$ is bounded.
While it's so trivial when normed spaces are considered, I cannot see why it holds in any topological linear space...
$\infty$ thanks!
Let $U$ be a neighbourhood of $0$. Choose a balanced neighbourhood $V$ of $0$ such that $V+V\subset U$. Pick $n_1,n_2 > 0$ such that $A\subset \lambda V$ for $\lvert\lambda\rvert \geqslant n_1$ and $B\subset \lambda V$ for $\lvert \lambda\rvert \geqslant n_2$. Let $n = \max\{n_1,n_2\}$. Then for $\lvert\lambda\rvert \geqslant n$ we have ...