Functional Analysis


By the end of this course, the student should have acquired knowledge, skills and competences in order to:

-Use fluently the elementary properties, results and procedures of functional analysis in Banach spaces;

-Apply the open mapping and the closed graph theorems, the Banach-Steinhaus and Hahn-Banach theorems.

General characterization





Responsible teacher

Cláudio António Raínha Aires Fernandes


Weekly - 4

Total - Available soon

Teaching language



Available soon


B. Rynne, M. Youngson, Análise Funcional Linear, IST Press, 2011.

B. Rynne, M. Youngson, Linear Functional Analysis, 2nd ed., Springer, 2007.

H. Brezis, Functional Analysis, Sobolev Spaces, and Partial Differential Equations. Springer, 2010.

W. Rudin, Functional Analysis. MacGraw-Hill, 2nd edition, 1991.

G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.


Teaching method

Available soon

Evaluation method

There are two tests and a grade for the participation of the students in the classes throughout the semester. Otherwise the student must succeed the final exam. More detailed rules are available in the Portuguese version.

Subject matter

  1.   Normed spaces: Examples of normed spaces. Finite dimensional normed spaces. Banach spaces. Separable spaces. Separability of spaces C[a,b] and Lp[a,b].
  2.  Hilbert spaces. Internal products. Bessel’s inequality and Parseval’s identity. Orthogonality. Orthogonal complements. Orthonormal bases in infinity dimensions. Separable Hilbert spaces. Fourier series.
  3.  Linear operators: Norm of a bounded linear operator. The space of bounded linear operators. Baire’s theorem. The open mapping and closed graph theorems. Invertible operators. Banach’s isomorphism theorem. The uniform boundedness principle. Banach-Steinhaus’ theorem.
  4.  Duality and the Hahn-Banach theorem: The dual space. Sublinear functionals and seminorms. The Hahn-Banach theorem in normed spaces. The general Hahn-Banach theorem. The bidual space. Reflexive spaces and dual operators. Projections and complemented subspaces. Weak and weak-* convergence.


Programs where the course is taught: