FUNCTIONAL ANALYSIS

Compulsory course. 5 credits.

Theory: 4 credits + Exercises : 1 credits

I. Overview

Metric spaces. Normed Spaces. Continuous linear mappings between two normed spaces and theorems about them. Hilbert spaces. Spectral theory of compact operators.

II. Prerequisites : Analysis 1, Analysis 2, Analysis 3.

III. Contents

Chapter 1 : Metric spaces : metric spaces, convergence, continuous mappings

Chapter 2 : Banach spaces : normed spaces, sequences and series in normed spaces

Chapter 3 : Linear continuous mappings : linear mappings, continuity of linear mappings. Theorems : Banach-Steinhaus, Open mappings, closed graphï, Hahn-Banach.

Chapter 4: Hilbert spaces : positive Hermitian bilinear forms, Schwartz inequality and Minkowski inequality. Hilbert spaces. Orthogonal systems. Riezs presentation theorem.

Chapter 5 : Spectrum of compact operators : Riezs theorem.

.

References

[1] J. Dieudonne; Foundations of modern analysis; Academic Press, New York, 1960.

[2] Duong Minh Duc, Functional analysis (Vietnamese); National University at Hochiminh City Publisher, Hochiminh City, 2000.

[3] S. Lang; Analysis; Addison–Wesley. Reading, 1969

[4] W. Rudin; Real and complex analysis; McGrowHill, New York, 1970.