Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1002	Special Topics in Real Analysis	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	1	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. SERB Ioan Valeriu, Ph.D., ivserb@math.ubbcluj.ro	
Aims	Our intention is to consolidate the knowlidges of the students to the study of some classes of convex  functions and their applications to the study of Orlicz spaces. The Orlicz spaces are more general than the usual Lebesque spaces, considered earlier. We consider also some results on the dual and the geometry of Orlicz spaces. To the seminar the students will be familiarized with some notions of the geometry of general Banach spaces.	
Content	Convex functions and their integral representation. The definition of N-functions. Properties of N-functions. A new definition for  N-functions. The composition and the complement (dual) of  N-functions. The Young inequality  for N-functions. Examples. Translations of the inequalities from N-functions to their duals. Comparability of two N-functions. Translations of comparability from N-functions to their duals. The main part of N-functions. Sufficient conditions  of equivalence of two functions. Construction of new classes of equivalence for N-functions. N-functions verifying the Delta 2 condition. The expression of Delta 2 condition in terms of dual N-functions. Orlicz classes of measurable functions. The relation between an Orlicz class with known Lebesque spaces. Integral inequality of Jensen. The structure of the Orlicz classes. The Orlicz space L(M) and the Orlicz norm. Complete Orlicz spaces. Holder inequality for functions in Orlicz spaces. Relation between Orlicz spaces L(M) and Orlicz classes when M verifies Delta 2 condition. Luxemburg norm on Orlicz spaces. The computation of Orlicz norm with new formulas. Separability of Orlicz spaces ; the dual of an Orlicz space. The general form for the continuous linear functionals on Orlicz spaces. The reflexivity of Orlicz spaces.	
Bibliography	1. I. Şerb, Capitole speciale de analiză reală. Curs scris pe calculator. Fasc 1-8., 87 pag.
2. M. A. Krasnoselskii, Ia. B. Rutiţkii, Funcţii convexe şi spaţii Orlicz, Moscova 1958 (l. rusă)
3. M. M. Rao, Z. D. Ren, Theory of Orlicz spaces, Pure and Applied Mathematics, Marcel Decker, New-York, Basel, Hong-Kong, 1991
4. D.V. Salehov, Despre norma funcţionalelor liniare în spaţiile Orlicz şi despre o caracteristică a spaţiilor Lp, Dokl. Acad. Nauk, SSSR, 111, 5 (1956) 948-950.
5. Z. Birnbaum, W. Orlicz, Uber die Veralgemeinerung des Begrifes der zveinander konjugierten Potenzen, Stud. Math. 3 (1931), 1-67.
6. I. Şerb, On the modulus of convexity of Lp spaces, Babeş-Bolyai University, Preprint 1, (1986) 175-187.
7. E. Popa, Culegere de probleme de analiză funcţională, Bucureşti, 1981.
8. Gh. Sireţki, Calcul diferenţial şi integral, vol I-II, Bucureşti, 1985
9. J. Diestel, Geometry of Banach spaces. Selected topic, Lecture Notes in Mathematics, Springer-Verlag, 1975
	
Assessment	The examination of the students: The first examination in the 8-th week of the semester from the material presented in the first 7 weeks. The second examination, is in the session of exam in January.
The both examinations consist in written papers. The arithmetic mean of the two papers is the final grade of the exam. If a student does not  participated at the first  date then he (or shi) can to obtain the final grade at the second date from the entire material. In the second session of exam the student obtains a single grade from the entire material. In the same conditions, a student can to obtain an improved  grade in the second session of exam.