Exercise 22, § 2

Let $E$ be a Banach space, $E'$ its dual and $M'$ a strongly closed subspace in $E'$ which is everywhere dense for the weak topology on $E'$. We say that $M$ is irreducible if there exists no vector subspace $N' \neq M'$ of $M'$ which is strongly closed and weakly everywhere dense in $E'$. a) Show that $M'$ is irreducible if and only if the orthogonal ${M'}^\circ$ of $M'$ in $E''$ is the topological complement of $E$ (for the strong topology of $E''$). Deduce that then ${M''}^{(1)} = E'$ (exerc. 21) and that $E$ is isomorphic to the strong dual of the space $M'$ endowed with the topology induced by the strong topology of $E'$. b) Show that $M'$ is irreducible if and only if the unit ball in $E$ is relatively compact for the topology $\sigma(E, M')$ (use exerc. 21, a)). c) For $E$ to be isomorphic to a strong dual of a Banach space (for the topological vector space structure), it is necessary and sufficient that there exist an irreducible subspace in $E'$. Deduce a new proof of the fact that the Banach space $\mathscr{K}(\mathbf{N})$ is not isomorphic to a strong dual of a Banach space (cf. IV, p. 55, exerc. 16, c)).