If $X$ and its dual $X^*$ have the Radon-Nikodym property then $X$ contains no copy of $c_0$.

Question

Let $X$ be a Banach space; the associated dual space is denoted by $X^*$ such that $X$ and its dual $X^*$ have the Radon-Nikodym property.

Why $X$ contains no copy of $c_0$?

An idea please.

Answer 1

Reference

Diestel, J.; Uhl, J. J. jun., Vector measures, Mathematical Surveys. No. 15. Providence, R.I.: American Mathematical Society (AMS). XIII, 322 p. $ 35.60 (1977). ZBL0369.46039.

Chapter III, Section 3, Theorem 2.

A Banach space has the Radon-Nikodym property if each of its closed separable linear subspaces has this property.
If a Banach space has the Radon-Nikodym property, so does each of its closed linear subspaces.

Let $X$ have the RNP. (No assumption about $X^*$.) Then any closed subspace $Y$ of $X$ also has RNP. Therefore, $Y$ is not isomorphic (= linearly hmeomorphic) to $c_0$.