Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ consider the space $C_\#(X,\mathbb{C})$ of finitely supported complex functions on $X$. Then $C_\#(I,\mathbb{C})$ is algebraically isomorphic to $V$ by definition of a Hamel basis, $$ \forall v\in V,\quad v=\sum_{e\in I}\lambda_e^v e,\quad I\ni e\mapsto\lambda_e^v\in\mathbb{C}. $$

**Question:** Does there exist a topology on $C_\#(X,\mathbb{C})$, such that $C_\#(I,\mathbb{C})$ is isomorphic to $V$ as topological vector spaces?

In finite dimensions $I$ is finite and hence discrete, so that all reasonable topologies on $C_\#(I,\mathbb{C})=\mathbb{C}^I$ coincide and give a positive answer. The question therefore pertains to infinite dimensions.

I am mostly interested in separable Hausdorff $V$. Thank you.