Post History

Let $F$ be either $R$ or $C$ and $U$ be open in $T$, that is, by definition of $T$, $U$ is open in $T_k$ for every $k$. Since, for every $k$, $(V,T_k)$ is a topological vector space, $+:V\times V\rightarrow V$ and $\cdot:F\times V\rightarrow V$ are continuous with respect to $T_k$. So, for every $k$, $+^{-1}(U)\in T_k$ and $\cdot^{-1}(U)\in T_k$, thus $+^{-1}(U)\in T$ and $\cdot^{-1}(U)\in T$. Therefore, $+$ and $\cdot$ are continuous with respect to $T$ which, in turn, implies that $(V,T)$ is a topological vector space.