# A quantity measuring the separability of Banach spaces retort

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## A quantity measuring the separability of Banach spaces

Let $$X$$ breathe a Banach space. It is unaffected for us to insert a quantity measuring the separability of sets as follows: for a subset $$A$$ of $$X$$, we clique

$$textrm{sep}(A)=inf{epsilon>0: Asubseteq K+epsilon B_{X}$$ for some countable subset $$K$$ of $$X}$$.

Clearly, $$A$$ is separable if and only if $$textrm{sep}(A)=0$$.

It is elementary that a Banach space $$X$$ is separable if $$X^{*}$$ is separable. My question is to give a quantitative version of this known outcome.

Question. $$textrm{sep}(B_{X})leq Ccdot textrm{sep}(B_{X^{*}})$$ for some universal ceaseless $$C$$ ?

Thank you.

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