Syllabus Math 547

Math 547. Banach Spaces Instructor Syllabus

1. Introduction [5 hours]
Basic properties, review of Hahn-Banach and Krein-Milman theorems, convexity, convex approximation, extreme points of certain finite dimensional Banach spaces.

2. Trace duality and certain norms on spaces of linear maps [15 hours]
Basics on tensor norms, duality form, minimal and maximal norm, local reflexivity, absolutely p-summing maps and factorization, duality for 2-summing maps, Lewis lemma, Auerbach basis, John’s theorem, further topics chosen by instructor such as duality for p-summing maps.  

3. Probabilistic techniques [10 hours]
Type and cotype, Banach-Mazur distance, applications to probabilistic techniques in Banach spaces, further topics chosen by instructor.

4. Further recommended topics [10 hours]
Optimization, Grothendieck's theorem, bases in Banach spaces, Kwapien's theorem, Slepian's lemma and Sudakov's Theorem, introduction to operator spaces.

5. Leeway and/or student presentations  [3 hours]

Recommended references: 

  • Banach Spaces for Analysts by P. Wojtaszczyk, Cambridge Studies in Advanced Mathematics, 1996
  • Summing and Nuclear Norms in Banach Space Theory by G. J. O. Jameson, London Mathematical Society Student Texts (No. 8), 1987
  • Factorization of Linear Operators and Geometry of Banach Spaces by G. Pisier, CBMS Regional Conference Series in Mathematics, 1986
  • Classical Banach Spaces by J. Lindenstrauss  and L. Tzafriri, Springer Verlag, 1996

Recommended assessment:

Regular homework (about 5 assignments) and a final project with presentation. 

Approved by the GAC December 2013.