Math 595 --Banach spaces
 
Time:  MWF 1.00-1.50pm MATH 595
Location: 
Altgeld Hall  347

Instructor:
Marius Junge     
Course email   Office hours: TBA
 
Grading:  50% HW, 50% Presentation

Course discription :  We we start with basic Banach space properties, in particular local properties,
                                     and, if time permits, consider some applications. I expect a presentation from every
                                     particpant.

Plan

Part I: Introduction
      1) Basic properties 
      2) Basis in infinite dimension
      4) p-summing maps
      5) John's theoreom

Part II: Tensor norms 
      1) Definition
      2) Largest and smallest tensor norm

           Operator ideal  
      3) Local  reflexivity
      4) p-summing maps and p-nuclear mapsI ,   II  
        III
       4) Kwapien's theorem

subspaces and quotients1,

subspaces and quotients2,

       More on Kwapien 

      5) Banach Lattices     
       6)  Injective and projective tensor products
       6)  Grothendieck's theorem
Part III: Type and Cotype
       1) Khintchine-Kahane
       2) Type and Cotype
       3) Kwapien's theorem
       4) TBA

Part IV: Operator spaces 

       1) Basic definition 
       2) tensor norms
       3) min and max spaces

Part V: Information theort and Quantum Information theory
      1) Basic definitions 
        2) Capacity and Entropy 
        3) Connection to p-summing maps


 HW1, HW2, HW3, HW4, HW5, HW6, HW7, HW8, HW9, HW10, HW11, HW12


Books:  I will use selected topics from different resources
Defant, Andreas(D-OLD); Floret, Klaus(D-OLD)
Tensor norms and operator ideals.
North-Holland Mathematics Studies, 176.
North-Holland Publishing Co., Amsterdam, 1993. xii+566 pp. ISBN 0-444-89091-2

Lindenstrauss, Joram; Tzafriri, Lior
Classical Banach spaces. I.
Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
Springer-Verlag, Berlin-New York, 1977. xiii+188 pp. ISBN 3-540-08072-4

Lindenstrauss, Joram; Tzafriri, Lior
Classical Banach spaces. II.
Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97.
Springer-Verlag, Berlin-New York, 1979. x+243 pp. ISBN 3-540-08888-1

Pisier, Gilles
Factorization theory  of linear operators and geometry of Banach spaces.
CBMS Regional Conference Series in Mathematics, 60.
Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. x+154 pp. ISBN 0-8218-0710-2

Pisier, Gilles(1-TXAM)
The volume of convex bodies and Banach space geometry.
Cambridge Tracts in Mathematics, 94.
Cambridge University Press, Cambridge, 1989. xvi+250 pp. ISBN 0-521-36465-5; 0-521-66635-X

Pisier, Gilles
Probabilistic methods in the geometry of Banach spaces. Probability and analysis (Varenna, 1985), 167--241,
Lecture Notes in Math., 1206, Springer, Berlin, 1986.

Pisier, Gilles(F-PARIS6-E)
Similarity problems and completely bounded maps.
Second, expanded edition. Includes the solution to "The Halmos problem". Lecture Notes in Mathematics, 1618.
Springer-Verlag, Berlin, 2001. viii+198 pp. ISBN 3-540-41524-6

Pisier, Gilles(1-TXAM)
The operator Hilbert space ${\rm OH}$, complex interpolation and tensor norms. (English. English summary)
Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103 pp.


Wojtaszczyk, P.(PL-PAN)
Banach spaces for analysts.
Cambridge Studies in Advanced Mathematics, 25.
Cambridge University Press, Cambridge, 1991. xiv+382 pp. ISBN 0-521-35618-0

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