"Babes-Bolyai" University of Cluj-Napoca
Faculty of Mathematics and Computer Science

Topics in the geometry of Banach spaces
Hours: C+S+L
Teaching Staff in Charge
Assoc.Prof. SERB Ioan Valeriu, Ph.D., ivserb@math.ubbcluj.ro
Appropriation of basic knowledges and methods of the modern field of geometry of Banach spaces is the main objective. Analytical and intuitive geometric methods are combined in the study of Banach spaces. The notions of uniform convexity and uniform smoothness are fundamental in this direction. Some estimates for the modulus of convexity and smoothness are given for particular Banach spaces. Applications in some area of functional analysis are also given.
1.Uniform convexity and uniform smoothness in Banach spaces: Strictly convex and smooth Banach spaces. Uniformly convex and uniformly smooth Banach spaces. Moduli of convexity and smoothness attached to a Banach space and their general properties. Liokoumovich example of a non-convex modulus of convexity of a uniformly convex Banach space.
2. The geometry of unit balls and the attached moduli of Banach spaces. Applications: Rectangular modulus and squareness modulus. Relations between them, properties of them. Characterization of Hilbert spaces in terms of some moduli. The Day-Nordlander theorem. Applications to unconditional convergence of series in Banach spaces. The theorems of Kadec and Lindenstrauss. Applications in the theory of fixed points for nonexpansive mappings. Spaces with normal structure. Estimates for moduli of some concrete spaces. The case of L_p spaces, p>1.
1. J.Diestel, Geometry of Banach spaces, Selected Topic, Lecture Notes in Mathematics 485, Springer, 1975.
2. R.Deville, G.Godefroy, V.Zizler, Smoothness and renormings in Banach spaces, Pitman 1992.
3. D.Amir, Characterizations of inner product spaces, Birkhauser Verlag, 1986.
4. I.Serb, On the modulus of convexity of Lp, spaces, Seminar on Functional Analysis and Numerical Methods, Preprint 1, 175-187 (1996).
5. I.Serb, Some estimates for the modulus of smoothness and convexity of a Banach space, Mathematica 34 (57) 1 (1992) 61-70.
6. I.Serb, On the behaviour of the tangential modulus of a Banach space I, II, Revue d'Analyse Numerique et de Theorie de l'Approx. 24 (1995),241-248 si Mathematica (Cluj) 38 (61) (1996),199-20.
7. I Serb, A Day-Nordlander theorem for the tangential modulus of a normed space, J.Math. Anal. Appl.. 209 (1997), 381-391.
8. I. Serb, Rectangular modulus, Birkhoff orthogonality and characterizations of inner product spaces, Comment. Math Univ. Carolin. 40 /1 (1999), 107-119.
9. I.Serb, Rectangular modulus and geometric properties of normed spaces, Math. Pannonica 10, 2 (1999),
10. I.Serb, Geometric properties of normed spaces and estimates for rectangular modulus, Math. Pannonica, 12,1 (2001), 27-38.