Cod: 22214
Department: DCET
Scientific area: Mathematics
Total working hours: 130
Total contact time: 20

Completing the student's basic knowledge in Linear Algebra and Mathematical Analysis, this curricular unit deepens their study of infinite dimensional normed spaces, with a special emphasis in Banach and Hilbert spaces. Linear operators and some of the fundamental theorems (Hahn-Banach, open mapping, closed graph, and uniform boundedness) are studied.

Banach spaces
Hilbert spaces
Linear operators

Upon completion of this curricular unit the student must know, and know how to apply, the fundamental properties and results of Banach and Hilbert spaces and of linear operators defined on those spaces.

  1. Differential calculus and the  Riemann integral (Review)
  2. The Lebesgue integral
  3. Normed and Banach spaces
  4. Inner product spaces and Hilbert spaces
  5. Linear operators in Banach spaces and some fundamental theorems
  6. Duality. Hahn-Banach theorem
  7. Linear operators in Hilbert spaces

Compulsory Reading 

B.P. Rynne, M.A. Youngson: Análise Funcional Linear, Coleção Ensino da Ciência e da Tecnologia, vol. 39, IST Press, Lisboa, 2011

 Optional Reading 

Erwin Kreyszig; Introductory Functional Analysis with Applications, Wiley Classics Library, Wiley, New York, 1989



Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (60%) and final evaluation (40%). Further information is detailed in the Learning Agreement of the course unit.

Pre-requisites: Students are recommended to have previously knowledge in the Linear Algebra curricular units, in the Elements of Infinitesimal Analysis curricular units and in the Topology curricular unit of the 1st cycle degree course on Mathematics and Applications (Universidade Aberta).