Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Analiză reală şi complexă (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MO261	Applied Functional Analysis	

Semester	Hours: C+S+L	Category	Status	
1	2+1+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
COBZAS Stefan	Ph.D.	Prof.	Mathematical Analysis and Optimization	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
To present some results  from  the geometric banach space theory with applications to the differentiability of convex functions and to holomorphic vector functions.	

IV. Course contents
1. Complements of Banach space theory
Support functionals -  applications to best approximation, Bishop-Phelps subreflexivity theorem and James  reflexivity theorem. Weakly compact sets - Eberlein-Shmulian  and James-Price theorems. Goldstein theorem on the weak star density of a Banach space in its bidual. Weakly compact operators and Gantmacher theorem.

2. Geometric theory of normed spaces
Strictly convex, locally uniformly convex and uniformly convex normed spaces.
The reflexivity of uniformly convex Banach spaces. The uniform convexity of the spaces Lp and their moduli of uniform convexity. Characterizations of extreme and smooth points of the unit balls in concrete Banach spaces. Banach spaces with normal structure and fixed point theorems for nonexpansive mappings.

3. Differential calculus in normed spaces 
Gateaux and Frechet differentials and relations between them. Calculus rules in normed spaces. The differential of the inverse mapping, the theorem of local inversion and the implicit function thoerem. Bilinear mappings anf second oreder differentials. Multilinear mappings and higher order differentials. Taylor formula. Conditioned extrema and Liusternik theorem.
Convex functions - continuity properties, Gateaux differentiability properties  (Mazur theorem), Frechet differentiability properties. Smoothness properties  of normed spaces- Gateaux and Frechet differentiabilty of the norm, smooth spaces, locally uniformly smooth and  uniformly smooth spaces,  Asplund spaces, examples.
Banach-valued functions with bounded variation and Riemann-Stieltjes integral. Holomorphic and analytic Banach-valued functions. The equivalence of weak and strong holomorphy. Applications to the spectral calculus of  linear operators.

4. Hahn-Banach type theorem for linear operators
Finite intersection property and Nachbin theorem on the norm-preserving extension of continuous linear operators. The characterization of Banach spaces having the finite intersection property. Kantorovich and Krein theorems on the extension of linear operators on ordered spaces.  Equivalence of Hahn-Banach theorem and  the  existence of supremum in  ordered vector spaces. 

	

V. Bibliography
1. DEVILLE R., GODEFROY G., ZIZLER V.: Smoothness and Renormings in Banach Spaces.New York: John Wiley, 1993.
2. FABIAN M. et al.: Functional Analysis and Infinite-Dimensional Geometry. Berlin - New York: Springer Verlag, 2001. 
3. GILES J. R.: Convex Analysis with Applications in the Differentiation of Convex Functions. London:  Pitman, 1982.
5. HOLMES R. B.: Geometric Functional Analysis. Berlin: Springer-Verlag, 1975.
6. KANTOROVICI  L. V., AKILOV G. P.: Analiza functionala. Bucuresti: Editura Stiintifica si Enciclopedica, 1986.
7. KUTATELADZE S. S.: Fundamentals of Functional Analysis. Dordrecht: Kluwer Academic Publishers, 1995. 
8. MUNTEAN I.: Analiza functionala. Capitole speciale. Cluj-Napoca: Universitatea Babes-Bolyai, 1990.
9. RUDIN W.: Functional Analysis. New York: McGraw Hill, 1973.
10. YOSIDA K.: Functional Analysis. Berlin: Springer-Verlag, 1995.

VI. Thematic of didactic activities per weeks

VII. Didactic methods used

VIII. Assessment
seminar activities 25%; the presentation of a topic at seminar 25%; written examination 50%	

IX. Additional references

	Dean,							Chair head,

	Date,							Course responsible,
								Prof. COBZAS Stefan, Ph.D.