Isomorphic Structure of Cesàro and Tandori Spaces

We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $\text{Ces}_{\infty }$ and its sequence counterpart $\text{ces}_{\infty }$ are isomorphic. This is rather surprising since $\text{Ces}_{\infty }$ (like Talagrand’s example) has no natural lattice predual. We prove that $\text{ces}_{\infty }$ is not isomorphic to $\ell _{\infty }$ nor is $\text{Ces}_{\infty }$ isomorphic to the Tandori space $\widetilde{L_{1}}$ with the norm $\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$ , where $\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$ . Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.