Consider $\Phi$ to be the space of sequences that have finitely many non-zero terms. The space is not closed in $\ell_1$, therefore $\ell_1/\Phi$ with the quotient topology is not Hausdorff, and so it cannot be metrizable. However, does there exist a metric on $\ell_1/\Phi$ that gives rise to a non-trivial topology. And even stronger question is, is $\ell_1/\Phi$ normable?
Quotient of $\ell_1$ by space of finite sequences
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banach-spaces