Subject: Applied Functional Analysis	Code: MO261	

Semester	 Hours: C+S+L	Type	 Section	
1	2+1+1	compulsory	Real and Compex Analysis - in English	

Teaching Staff in Charge	Prof. COBZAS Stefan, Ph.D., scobzas@math.ubbcluj.ro	
Aims	To present some results  from  the geometric banach space theory with applications to the differentiability of convex functions and to holomorphic vector functions	
Content	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. 

	
Selective References	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.	
Assessment	Exam