Biorthogonal systems in Banach spaces C(K)

Author(s)

Clayton Suguio Hida

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2014-08-07

Examining board members

Christina Brech; Jorge Lopez Abad; Piotr Boleslaw Koszmider

• Advisor: Christina Brech

Abstract

The main purpose of this work is to apply elements of set theory to the study of biorthogonal systems in Banach spaces. Initially, we study Markushevic\'s Theorem, which ensures that every separable Banach space has a countable biorthogonal system. With this result, we focus our attention to the study of nonseparable Banach spaces, more especifically, we study the existence of uncountable biorthogonal systems in Banach spaces of the form C(K), with K a nonmetrizable compact Hausdorff space. In this direction, we study two theorems of S. Todorcevic. The first one gives us sufficient conditions that a compact Hausdorff space K must satisfy in order to get that the respective Banach space C(K) has an uncountable biorthogonal system. The second one tells us that under Martin\'s Axiom, every nonseparable Banach space of the form C(K) has an uncountable biorthogonal system. Next, we consider some cardinal functions defined by P. Koszmider for Banach spaces, related with biorthogonal systems, and we study its relations with well - known cardinal functions. In particular, we obtain an original result relating the weight of a compact Hausdorff space K to the size of certain biorthogonal systems in C(K), generalizing a result of S. Todorcevic for Boolean algebras. Finally, we construct an Ostaszewski space K using the Diamond Principle. The compact space K is a scattered nonmetrizable Hausdorff space such that all its finite powers are hereditarily separable. This space is a consistent example of a nonmetrizable compact Hausdorff space such that the respective Banach space C(K) does not have an uncountable biorthogonal system. (AU)

• FAPESP's process: 11/15129-3 - Partial orders and its applications to the geometry of Banach spaces
• Grantee: Clayton Suguio Hida
• Support Opportunities: Scholarships in Brazil - Master