Banach spaces 1000-1M25PB

As part of this course, students will become familiar with selected
topics in the geometry of Banach spaces and operator theory. The aim of
the course is to acquaint students with a range of tools in the theory
of Banach spaces. The following topics will be discussed:

Basic properties of sequence spaces (about 3 lectures)

Properties of Schauder bases

Examples of Schauder bases

Subspaces of lp and c0 spaces

Block bases

The Haar basis in L1 space

Rademacher type and cotype (about 3–4 lectures)

The Khinchin–Kahane inequality

Basic properties of type and cotype

Type and cotype of Lp spaces

The Kadets–Pełczyński theorem

The Kwapień–Maurey factorization theorem for spaces of type 2

Martingale type and its relation to the superreflexivity of Banach
spaces

Pełczyński’s decomposition principle (2 lectures)

The isomorphism between l∞ and L∞, Sobczyk’s theorem

Classification of H1(Fn) spaces

Auerbach bases, the local reflexivity principle, Pełczyński’s
theorem on (1+ϵ)-Markushevich bases (1 lecture)

The Dunford–Pettis property for L1, C(K), and C1 spaces (1 lecture)

p-absolutely summing operators (3–4 lectures)

p-summing operators in Hilbert spaces

Pietsch’s theorem

Grothendieck’s theorem

Bennett–Carl theorem

If one lecture remains, I will present Bourgain’s result on polydisc
algebras.