Functional analysis: fundamentals

Basic information

Workload: 

45h

Prerequisite: 

Does not exist

Syllabus: 

Metric spaces, standardized and with internal product. Continuous functions in metric spaces, completeness, Banach Fixed Point Theorem, compactness, density, separability, continuous applications between metric spaces, Tietze extension theorem, Arzelà-Ascoli theorem, Stone-Weierstrass theorem. Topological spaces. Banach spaces, limited linear functionalities, convexity, the Hahn-Banach theorem. Hilbert spaces, orthogonality, Projection Theorem, Fourier analysis, Riesz Representation Theorem. Applications and examples.

Associated lines of research:

Bibliography

Mandatory: 

· Bachman, Narici (2000). Functional Analysis. Dover.
· Saxe (2002). Beginning Functional Analysis. Springer.
· Kolmogorov, Fomin (1982). Elements of Theory of Functions and Functional Analysis. MIR.

Complementary: 

· Bollobás (1999). Linear Analysis. Cambridge.
· Friedman (1982). Foundations of Modern Analysis. Dover.
· Oliveira (2015). Introduction to Functional Analysis. IMPA.
· Bobrowski (2005). Functional Analysis for Probability and Stochastic Processes: An Introduction. Cambridge.
· Atkinson, Han (2009). Theoretical Numerical Analysis: A Functional Analysis Framework (TAM). Springer.
· Lages (2007). Metric Spaces. IMPA.