Exercise 17, § 2

Let E be a non-reflexive Banach space, E' its strong dual, E'' the strong dual of E', E''' the strong dual of E'' and E^{IV} the strong dual of E'''. a) Show that, in E''', E' and the subspace $E^\circ$ orthogonal to E (when E is considered as a subspace of E'') are topological complements, and that the projection from E''' onto E' for this decomposition is a continuous linear mapping of norm 1. Comparing with exerc. 16, c), deduce that the Banach space $\mathscr{K}(A)$ is not isomorphic (as a topological vector space) to the strong dual of any Banach space. b) Show that E^{IV} is the topological direct sum of ${E'}^\circ$ and E'', and also of ${E'}^\circ$ and ${E''}^\circ$; we have $E'' \cap {E''}^\circ = E$. Let $v$ be the linear mapping from E^{IV} onto itself which is identity on ${E'}^\circ$, and on E'', is the projection from E'' onto ${E''}^\circ$ parallel to ${E'}^\circ$; show that $v$ is an isometry, but is not continuous for the topology $\sigma(E^{IV}, E''')$.