From \(\ell^p\)-spaces to Gowers-Maurey spaces (Q2721278)

This short beautiful expository paper is an introduction to the constructions by Gowers and by Gowers and Maurey and to the resulting advances in Banach space theory in the last decade. These include the solution of several famous problems stated by Banach. NEWLINENEWLINENEWLINEThe paper starts with the basic definitions of a Banach space, and the classical spaces, \(c_0\), \(\ell_p\), \(C[0,1],\) and \(L_p[0,1]\). Discussing their properties permits the author to motivate questions about Schauder bases, unconditional bases, prime spaces, and reflexivity. The author progresses through landmark results by James, Pelczynski, Tsirelson, Figiel and Johnson, Enflo, and Szankowski. These motivate questions on distortable spaces, subspaces isomorphic to the ambient space, and hereditarily indecomposable spaces. The paper then lists some of the remarkable consequences of the Gowers and Maurey construction, before giving the definition of the space they defined. The paper concludes with Gowers' dichotomy theorem and his verification of the Banach conjecture that \(\ell_2\) is the only space isomorphic to each of its subspaces. NEWLINENEWLINENEWLINEDefinitions, but not proofs, are given as the exposition flows smoothly from the very basics to the recent Field Prize winning results. At least a dozen significant results are given context in this brief but pithy presentation.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00027].