It is shown that for every separable Banach space $X$ the cardinality of $X$ is at most cantinuum; i. e. $card(X)\leq c= \aleph_1.$ Is the converse of this statement also true?
2026-04-16 15:21:03.1776352863
The relation between the separability and cardinality of a Banach space
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No, for example $\ell^\infty$, the inseparable space of bounded sequences, is a subset of the set $\Bbb{R}^\Bbb{N}$, the set of all real sequences, which has cardinality $\mathfrak{c}$.