Uniform structure of Banach spaces

Author(s)

João Gabriel Vitor de Carvalho

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2021-03-19

Examining board members

Wilson Albeiro Cuellar Carrera; Willian Hans Goes Corrêa; Thiago Grando

• Advisor: Wilson Albeiro Cuellar Carrera

Abstract

A common problem in mathematics is to determine when two objects have the same structure, for instance, determine when two topological spaces are homeomorphic. In Functional Analysis, it is known that two isomorphic Banach spaces have the same structure as topological vector spaces, but it is possible that others classes of functions preserve this structure. A strong result in this way is the classic Mazur-Ulam theorem, that asserts that a surjective isometry between real Banach spaces is an affine function, hence isometries also preserve the linear structure of real Banach spaces. In chapter 3 we deal with Lipschitz isomorphisms and embeddings. Concepts such as Haar null sets, Gâteaux differentiability and Radon-Nikodým property are developed. We show a result of Heinrich and Mankiewicz, that says that if X and Y are Banach spaces and Y has the Radon-Nikodým property, then every Lipschtiz embedding f: X \\to Y can be linearized through the Gâteaux derivative. This result will make way to many others results concerning Lipschitz isomorphisms and stable properties under Lipschitz isomorphisms. It is shown, for instance, that for 1 < p < \\infty, every Banach space Lipschitz isomorphic to a L_ space is also linearly isomorphic to L_ . In chapter 4, we deal with uniform homeomorphisms. Results such as the Gorelik principle and results from Ramsey theory enable bounds related to the uniform homeomorphisms that will restrict when two Banach spaces can be uniform homeomorphic. Then, we can prove the classic theorem of Johnson, Lindenstrauss and Schechtman concerning the uniform structure of the spaces \\ell_\'s: if 1 < p < \\infty and X is a Banach space uniformly homeomorphic to \\ell_, then X is linearly isomorphic to \\ell_. In the end, it is proved the same for the spaces \\ell_\\oplus\\ell_, where 1 < p < q < 2 or 2 < p < q < \\infty and afterwards for the spaces \\ell_\\oplus\\ell_, where 1 < p < 2 < q < \\infty. The first one was established by Johnson, Lindenstrauss and Schechtman and the second one was established by Kalton and Randrianarivony. (AU)

• FAPESP's process: 18/25569-0 - Uniform structure of Banach spaces
• Grantee: João Gabriel Vitor de Carvalho
• Support Opportunities: Scholarships in Brazil - Master