= Countable annihilation sequence in an irreducible Banach module
{title2=$\pi(a_m\cdots a_1)\xi_n=0\ (m>n)$}
For countably many <linearly independent> <vectors> in an infinite-dimensional irreducible complex Banach module, there are elements $a_n$ satisfying the displayed annihilation conditions while the tail images $\{\pi(a_n\cdots a_1)\xi_j:j\ge n\}$ remain <linearly independent>. Use the <quotient norm on an irreducible Banach module> to make each finite annihilator a <closed> Banach subspace. The <Jacobson density theorem> makes each condition of independence of finitely many tail images <open> and dense there; the <Baire category theorem> intersects these countably many conditions. This is stronger than prescribing only finitely many images once.
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