= Solution
The <algebraic dual of a direct sum> is a product, rather than another <direct sum>. The map
$$
R:V^*\longrightarrow\prod_{i\ge1}V_i^*,\qquad\ell\longmapsto(\ell|_{V_i})_i
$$
is injective, since the summands span $V$. Given any family $(\ell_i)_i$, define $\ell((v_i)_i)=\sum_i\ell_i(v_i)$. This sum is finite for each vector, regardless of whether the family of functionals has finite support. It defines a <linear functional> and is inverse to $R$. Hence this is a natural vector-space isomorphism.
The <algebraic contragredient representation> is $(\pi^*(g)\ell)(v)=\ell(\pi(g^{-1})v)$. On the $i$th coordinate,
$$
(\pi^*(g)\ell)_i(v_i)=\ell_i(\chi_i(g)^{-1}v_i)=\chi_i(g)^{-1}\ell_i(v_i).
$$
Thus $R$ intertwines the actions and gives
$$
\boxed{(V^*,\pi^*)\cong\prod_{i\ge1}(V_i^*,\chi_i^{-1}).}
$$
No smoothness restriction has yet been imposed on that product.
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