Let p 2 (0; 1), let v be a weight on (0; 1) and let p (v) be the classical Lorentz space, determined by the norm kfk p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. When p 2 (1; 1), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if p (v) = ? p (v), where kfk ? p (v) := (R 1 0 (f (t)) p v(t) dt) 1=p. We may thus conc...

The title above is wrong, because the strong dual of a Banach space is too strong to assert that the natural correspondence between a space and its bidual is an isomorphism. This, from a categorical point of view, is indeed the right duality concept because it yields a self adjoint dualisation functor. However, for many applications the non–reflexiveness problem can be solved by replacing the n...