Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which:
a) makes the $f_\alpha$'s continuous
b) makes vector addition continuous
c) makes scalar multiplication continuous
When is condition a) enough to imply conditions b) and c)?
In other words:
Consider the sets: $\{x \in V\rvert x\in f_\alpha^{-1}(B) \}$ with $\alpha$ varying over $A$ and $B$ over all the open balls in $\mathbb{C}$. When do these sets form a subbase for a topology which satisfy b, and c? (Where by construction a is already satisfied)
Edit: I'm not even sure that what I said below works...
For instance I know it works when $f_\alpha$ are linear, and a) $\implies$ b) when $f_\alpha$ are seminorms. Are there more general conditions?