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Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i > 0}|x_i|$. I want to prove that $m \not \cong \ell$. But have no idea how to do it. They both not compact(since not sequentially compact) and i don't know other invariants for such spaces.

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Hint: Try showing that $\ell$ is not separable under the sup-norm, but it is separable under the other norm. And recall that continuous images of separable spaces are separable.

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