Let $A \subseteq \mathbb{C}$ be path-connected and let $V_A$ be the $\mathbb{C}$-vector space of continuous maps $A \to \mathbb{C}$. Let $W_A$ be the set of piecewise $C^1$ maps $[0,1] \to A$. Then we have a map $$I : W_A \to V_A^\star$$ ($V_A^\star$ denotes the dual space of $V_A$) given by $$I(\gamma) = f \mapsto \int_\gamma f(z) \, \text{d} z.$$
What (nice) topology should we choose on $V_A^\star$ and $W_A$ so that $I$ is continuous? Really, we just need to pick a topology on $V_A^\star$ and then we let $W_A$ have the initial topology with respect to $I$.
One approach would be defining a topology on $V_A$ such that $I(\gamma)$ is continuous for all $\gamma \in W_A$ (e.g. the initial topology with respect to $\{I(\gamma) : \gamma \in W_A\}$ obviously works). Then, $I$ maps into the topological dual, and so we just need to choose a nice topology on the topological dual (e.g. the strong topology or weak topology). But I'm not sure which topologies will yield nice properties, and maybe the best choice of topology isn't constructible this way at all!
Any help is appreciated! I don't know much about complex analysis or topology so I don't know where to approach this problem from; I just got curious about it today in class.