10 - Back to Banach Space Theory

Summary

The final chapter of the book returns to the place the journey started: classical Banach space theory, with a twist. We can now provide solutions, or at least a better understanding, for a number of open problems. Among the topics covered, the reader will encounter vector-valued forms of Sobczyk’s theorem, isomorphically polyhedral $\mathscr L_\infty$-spaces, Lipschitz and uniformly homeomorphic $\mathscr L_\infty$-spaces, properties of kernels of quotient operators from $\mathscr L_1$-spaces, sophisticated 3-space problems, the extension of $\mathscr L_\infty$-valued operators, Kadec spaces, Kalton-Peck spaces and, at last, the space $Z_2$. All these topics can be easily considered as part of classical Banach space theory, even if the techniques we employ involve most of the machinery developed throughout the book.