Exercise 12, § 1

Let E be a reflexive Banach space of infinite dimension. Let $u\in \mathscr{L}^c(E)$ be a compact endomorphism. Suppose that $u$ is not of finite rank. Let $E_{\sigma}$ be the space E endowed with the weakened topology.

Now prove that the linear mapping $^tu$ is a compact endomorphism of $(E_{\sigma})'$.