| Elemente de geometria spaţiilor Banach | Topics in the geometry of Banach spaces |
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(Mathematics) |
| Cadre didactice indrumatoare | Teaching Staff in Charge |
| Conf. Dr. ŞERB Ioan Valeriu, ivserb@math.ubbcluj.ro |
| Obiective | Aims |
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Se urmareste insusirea unor cunostinte si metode de lucru din domeniul modern al geometriei spatiilor Banach. Se impletesc metodele analitice cu cele intuitive, de natura geometrica, pentru studiul spatiilor Banach. Notiunile de uniform convexitate si uniform netezime sunt fundamentale in acest context. Se dau evaluari ale modulelor de convexitate si de netezime pentru spatii concrete. Se dau aplicatii ale metodelor geometrice considerate. |
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. |
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1. Uniform convexitate si uniform netezime in spatii Banach: Spatii Banach strict convexe si spatii Banach netede. Spatii Banach uniform convexe si uniform netede. Modulele de convexitate si netezime. Proprietati ale modulelor de convexitate si de netezime ale spatiilor Banach. Exemplul lui Liokoumovich de modul de convexitate neconvex.
2. Geometria bilelor unitate si modulele atasate spatiilor Banach. Aplicatii: Modulul de patraticitate, modulul rectangular, alte module. Relatii intre module. Proprietatile acestor module. Caracterizarea spatiilor Hilbert cu ajutorul modulelor. Teoreme de tip Day-Nordlander. Aplicatii privind convergenta neconditionata a seriilor dintr-un spatiu Banach. Teoremele lui Kadec si Lindenstrauss. Aplicatii in teoria punctului fix. Spatii cu structura normala. Determinarea unor module pentru spatii concrete. Cazul spatiilor Hilbert. Cazul spatiilor Lp, p > 1. |
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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), 211-222. 10. I.Serb, Geometric properties of normed spaces and estimates for rectangular modulus, Math. Pannonica, 12,1 (2001), 27-38. |
| Evaluare | Assessment |
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Examen. |
Exam. |