Applied functional analysis 
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Teaching Staff in Charge 

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, BishopPhelps subreflexivity theorem and James reflexivity theorem. Weakly compact sets  EberleinShmulian and JamesPrice 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. Banachvalued functions with bounded variation and RiemannStieltjes integral. Holomorphic and analytic Banachvalued functions. The equivalence of weak and strong holomorphy. Applications to the spectral calculus of linear operators. 4. HahnBanach type theorem for linear operators Finite intersection property and Nachbin theorem on the normpreserving 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 HahnBanach theorem and the existence of supremum in ordered vector spaces. 
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 InfiniteDimensional 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: SpringerVerlag, 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. ClujNapoca: Universitatea BabesBolyai, 1990. 9. RUDIN W.: Functional Analysis. New York: McGraw Hill, 1973. 10. YOSIDA K.: Functional Analysis. Berlin: SpringerVerlag, 1995. 
Assessment 
Exam. 