# banach spaces – On weakly precompact sets

Recall that a bounded subset $$A$$ of a Banach space $$X$$ is said to be weakly precompact if every sequence in $$A$$ admits a weakly Cauchy subsequence. Rosenthal’s $$l_{1}$$-theorem states that a bounded subset $$A$$ is weakly precompact if and only if it contains no $$l_{1}$$-sequence. For a bounded subset $$A$$ of $$X$$, we let $$operatorname{wpc}_{X}(A)=inf{epsilon>0:Asubseteq K_{epsilon}+epsilon B_{X}},$$ where $$K_{epsilon}$$ is weakly precompact.

It is easy to see that $$operatorname{wpc}_{X}(A)=0$$ if and only if $$A$$ is weakly precompact. Clearly, every weakly Cauchy sequence is weakly precompact. Next I want to know how to quantify this simple fact.

Let $$(x_{n})_{n}$$ be a bounded sequence of $$X$$. We let $$delta((x_{n})_{n})=sup_{x^{*}in B_{X^{*}}}inf_{n}sup_{k,lgeq n}|langle x^{*},x_{k}-x_{l}rangle|.$$ Clearly, $$delta((x_{n})_{n})=0$$ if and only if $$(x_{n})_{n}$$ is weakly Cauchy. It is easy to see that $$delta((x_{n})_{n})$$ is the diameter of the set of all $$weak^{*}$$-cluster points of $$(x_{n})_{n}$$ in $$X^{**}$$.

Question. $$operatorname{wpc}_{X}((x_{n})_{n})leq delta((x_{n})_{n})$$

Thank you !