Exercise 23, § 2

With the same notations as in exerc. 22, assume that $M'$ is irreducible. a) In order that the canonical mapping from $E$ into $E''/{M'}^\circ$, which is the restriction of the canonical mapping $E'' \to E''/{M'}^\circ$ be a Banach space isometry, it is necessary and sufficient that the characteristic (IV, p. 55, exerc. 21) of $M'$ is equal to 1. Then, $M'$ endowed with the norm induced by that of $E'$ is said to be the predual of $E$ and $E$ can be canonically identified (with its norm) with the dual of the Banach space $M'$. b) For every vector subspace $F$ of $E$ which is closed for $\sigma(E, M')$, show that the canonical image of $M'$ in the dual $F'$ of $F$, identified with $E'/F^\circ$, is a predual of $F$.