Math 595 --Banach spaces
Time:  MWF 11-11.50pm
Altgeld Hall  441

Marius Junge     
Course email   Office hours: TBA
Grading:  50% HW, 50% Presentation (we will have more time)

Course discription :

Part I: Introduction
      1) Introduction
      2) Hahn-Banach
      3) Finite dimensional spaces and Ellipsoids
      4) p-summing maps and trace duality 1
      5) John's theoreom

Part II: Tensor norms 
      1) Definition
      2) Largest and smalles tensor norm
      3) injective and surjective tensor norms
      4) p-summing maps and p-nuclear maps
      5) Examples of trace duality
      6) Abstract factorization 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

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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