= Conductors force finite support in a smooth dual
If one-dimensional smooth <linear characters> of a local multiplicative group have conductors tending to infinity, their <direct sum> can be admissible. Its algebraic dual is the product of inverse <linear characters>, but a <smooth vector> in this product has only finitely many nonzero coordinates. A common open stabilizer contains one fixed principal-unit subgroup, on which every sufficiently ramified <linear character> is nontrivial. Thus the smooth dual is again a <direct sum>.
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