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+0		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 theory of Banach spaces 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 theorem. Bilinear mappings and second order 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 differentiability of the norm, smooth spaces, locally uniformly smooth and uniformly smooth spaces, Asplund spaces, examples.
Banach-valued functions with bounded variation and the Riemann-Stieltjes integral. Holomorphic and analytic Banach space-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
The binary 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
C.1 Support functionals, Bishop-Phelps subdifferentiability theorem, James characterization of reflexivity.
S.1 Support functionals in concrete Banach spaces, characterization of best approximation in terms of the support functionals.
C.2 Weak compactness in Banach spaces –Eberlein-Shmulian theorem and   James-Price theorem on the characterization of weak compactness. 
S.2 Examples of compact and weakly compact subsets in concrete Banach spaces.
C.3  Goldstein’s theorem on the weak density of a Banach space in its bidual. Weakly compact operators – Gantmacher’s theorem.
S.3 Unconditionally convergent series in Banach spaces – Orlicz theorem.
C.4 Strictly convex normed spaces – definition, characterizations and applications to the uniqueness of best approximation.
C.5 Uniformly convex (UC)  normed spaces– definition, the  modulus of uniform convexity, sequential characterization, the reflexivity of UC Banach spaces. Locally uniformly convex spaces (LUC)-definition, sequential characterization, the Kadec-Klee property of LUC spaces.
S.5 Examples of concrete UC and LUC Banach spaces – Hilbert spaces, Lp-spaces.
C.6 Normal structure and fixed points for nonexpansive mappings.
S.6 Examples related to Banach contraction principle.
C.7 Frechet (F) and Gateaux (G)  differentiability of  functions defined on open subsets of normed spaces –definitions, elementary properties, relations between G- and F-diffrentiabilty,  calculus rules.
S.7 The Frechet differentiability of convex functions of n variables.
C.8  Mean value theorems and applications – relations between Gateaux and Frechet differentiability, partial differentials.
S.8 Vector functions of bounded variation and the Riemann-Stieltjes integral for vector functions.
C.9 The differentiability of the inverse mapping, the inverse mapping theorem and the implicit mapping theorem.
S.9 Conditional extrema and Liusternik’s theorem.
C.10 Multilinear mappings and higher order differentials. Taylor’s formula.
S.10 Vector holomorphic functions – the equivalence between weak and strong holomorphy.
C.11 Gateaux differentiability of convex functions on normed spaces. Mazur’s theorem on the generic G-differentiability of convex functions on separable Banach spaces.
S.11 Baire spaces and Baire theorem. 
C.12 The generic F-differentiability of convex functions on Banach spaces with separable dual, Asplund spaces.
S.12 Differentiability of the norm in concrete Banach spaces.
C.13 Hahn-Banach type theorems for linear operators – the binary intersection property and norm preserving extensions of continuous linear operators, the characterization of Banach spaces with the binary intersection property.
S.13 Differentiability properties of the inversion mapping in Banach algebras. The spectrum of an element – compactness, nonenptiness. 
C.14 Hahn-Banach type theorems for linear operators on ordered Banach spaces – the sup-property in ordered Banach spaces, Kantorovich and Krein theorems.
S.14 Applications to the existence of some finitely additive measures.	

VII. Didactic methods used

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

IX. Additional references

Dean: Prof. Leon Tambulea, Ph.D.     			Chair Head: Prof. Stefan Cobzas, Ph.D 

Date, september 15, 2008					Course responsible,
								Prof. COBZAS Stefan, Ph.D.