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12778 Functional Analysis Supplements - Five-year degree in Mathematics

Center

Faculty of Mathematics

Departament

Mathematical Analysis

Lecturers in charge

F8060 - ENRIQUE LLORENS FUSTER

Met. Docent

Met. Avaluació

Final examination - -

Bibliografia

BRÉZIS, H. Análisis Funcional. Alianza Universidad, 1983
CONWAY, J.B. A Course in Functional Analysis. Springer-Berlag, 1985
DUNFORD, J.T., SCHWARTZ J. Linear Operator I. Interscience Publisher, 1958
KOLMOGOROV, A.H., FOMIN, S.V. Elemantos de la Teoría de funciones y del Análisis Funcional. Mir, 1975
TAYLOR, A.E., LAY, D.C. Introduction to Functional Analysis. John Wiley and Sons, 1980

Continguts

Goals
This course pretends to be a complentary part of the Functional Analysis course given in previousyears. We introduce the basic tools of Functional Analysis such as Banach, Hilbert spaces and spectral theory forself-adjoint compact operators. We also work on the three basic foundations of Functional Analysis:Hahn-Banach's Theorem, Banach-Steinhaus' Theorem and the Closed Graph Theorem, also dealing with some oftheir standard consequences.

Theory programme
Chapter 1. Hahn-Banach's Theorem Analytic version of the Hahn-Banach theorem and its consequences. Convex sets. Geometric version of the Hahn-Banach theorem. Separation theorems.Chapter 2. The Banach-Steinhaus and closed graph theorems Baire's theorem. The theorem of Banach-Steinhaus. The open mapping and closed graph theorems.Chapter 3. The weak topology. Reflexive spaces. The weak topology of a normed space. Weak convergence. The weak* topology. Alaoglu's theorem. Reflexive spaces.