= Solution
Let $\ell=(\ell_i)_i$ be a <smooth vector> in the product description of the <algebraic contragredient representation>. Its open stabilizer contains $U_m$ for some $m$. For every $i\ge m$, choose $u_i\in U_i\subseteq U_m$ with $\chi_i(u_i)\ne1$. Invariance of $\ell$ at its $i$th coordinate gives
$$
\chi_i(u_i)^{-1}\ell_i=\ell_i,
$$
forcing $\ell_i=0$. Hence a smooth functional has finite support, contained among $i<m$.
Conversely a finitely supported family has indices bounded by some $N$. Each inverse <linear character> has the same kernel as its original <linear character>, and all those kernels contain $U_{N+1}$. Thus this subgroup fixes the family. The <smooth dual> is therefore
$$
\boxed{V^\vee=\bigoplus_{i\ge1}V_i^*.}
$$
This is the principle that <conductors force finite support in a smooth dual>. For instance a product vector with every coordinate nonzero is a perfectly valid algebraic functional on the original <direct sum>, but is not smooth.
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