Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector space generated by $ \bigcup_{ \alpha \in A} F_{\alpha}$. Show that $ (V,\mathcal F)$ is also a topological vector space.
I started by showing that $+ : V \times V \to V$ is continuous in $\mathcal F$. But I turned out suspiciously did not use the fact it is addition, i.e. I showed that any function $g : V \times V \to V$ that is continuous in all $ \mathcal F_ \alpha$ is also continuous in $\mathcal F$. I don't know if this conclusion is even true.
Anyone help me on this problem. Thanks a lot.
