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A bounded subset $A$ of $X^*$ is called an L-set if each weakly null sequence $(x_n)$ in $X$ tends to zero uniformly on $A$.

The space $X$ has the Reciprocal Dunford-Pettis property if every L-subset of $X^*$ is relatively weakly compact; equivalently, every completely continuous operator from $X$ to any Banach space $Y$ is weakly compact.

I would like to find a Banach space $X$ so that every L-subset of $X^*$ is weakly precompact (that is every sequence in the set has a weakly Cauchy subsequence) but does not have the Reciprocal Dunford-Pettis property.

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