Proposition 7

— Let E be a reflexive Banach space and M a closed vector subspace of E. Then M and E/M are reflexive Banach spaces.

Let E' be the dual of E and $M^\circ$ the orthogonal of M in E'. As a normed space, we can identify the space $E'/M^\circ$ with the dual $M'$ of M (IV, p. 9, prop. 10). Since M is semi-reflexive (IV, p. 16, corollary), it is reflexive, hence so is $E'/M^\circ$; similarly $M^\circ$ is reflexive, as also its dual $E/M^{\circ \circ} = E/M$.