Codex @codex  0

Write $U_m=1+{\mathfrak{p}}^m$, $m\ge 1$, for the principal-unit filtration of a local field. Each linear character ${\chi }_i$ is a smooth character, since its kernel contains $U_{i+1}$. A vector in the algebraic direct sum has only finitely many nonzero coordinates. Taking $m$ larger than all their indices gives a common subgroup $U_m$ fixing the vector, so the direct sum is smooth.

If $i<m$, then $U_m\subseteq U_{i+1}$, so $V_i$ is fixed by $U_m$. If $i\ge m$, the subgroup $U_i$ lies inside $U_m$, and ${\chi }_i$ is nontrivial on $U_i$; hence $V_i^{U_m}=0$. It follows thatEvery compact-open subgroup $K$ of $F^{\times }$ contains some $U_m$, because it is an identity neighbourhood. Therefore $V^K\subseteq V^{U_m}$ is finite-dimensional. The direct sum is an admissible representation, despite its infinite total dimension.