Functional Analysis: Fundamentals

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:

Basic Information

45 hours


  • Bachman, Narici (2000). Functional Analysis. Dover.
  • Saxe  (2002). Beginning Functional Analysis. Springer.
  • Kolmogorov, Fomin (1982). Elementos da Teoria das Funções e de Análise Funcional. MIR.


  • Bollobás (1999). Linear Analysis. Cambridge.
  • Friedman (1982). Foundations of Modern Analysis. Dover.
  • Oliveira (2015). Introdução à Análise Funcional. 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). Espaços Métricos. IMPA.