This is a common error made by mature mathematicians in many books and papers in analysis, especially in differential equations: If X $X$ is a closed subspace of a Banach space Y, $Y$, then the Y^* $Y^*$ (the dual of Y) $Y$) is isomorphic to a subspace of X^* $X^*$ (the dual of X). $X$). It is false (of course) since Euclidian space R $\mathbb R$ is a subspace of R^2$\mathbb R^2$, yet the dual of R^2=R^2 $\mathbb R^2=\mathbb R^2$ is not isomorphic to a subspace of the dual of R=R$\mathbb R=\mathbb R$. I guess, sometimes they really, really want it to be true. Cheers Boris