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Sep 11, 2021 at 9:48 comment added Dongyang Chen @OnurOktay I add something to the question.
Sep 11, 2021 at 9:43 history edited Dongyang Chen CC BY-SA 4.0
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Sep 11, 2021 at 9:32 comment added Dongyang Chen Thanks, Onur. The $\delta((x_{n})_{n})=0$ if and only if $(x_{n})_{n}$ is weakly Cauchy.
Sep 11, 2021 at 8:27 comment added Onur Oktay The paper doi.org/10.1016/j.aim.2012.10.019 provides a comprehensive background for this problem. Especially Theorems 5.4, 5.5 and Examples 5.10, 10.1 are instructive.
Sep 10, 2021 at 15:40 comment added Onur Oktay There might be a typo above. Isn't $\inf_n\sup_{k,l\geq n} |\langle x^*,x_k-x_l\rangle| = 0$ for each $x^*\in X^*$ and weakly null sequence $(x_n)$ in any Banach space $X$?
Sep 10, 2021 at 9:16 history edited YCor
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Sep 10, 2021 at 3:08 history asked Dongyang Chen CC BY-SA 4.0