Functional Analysis


Complete metric spaces. Fixed point theorem. Theorem of embedded balls. Baire theorem. Compactness: equivalent definitions. Arzela-Ascoli theorem. Normed linear spaces and applications: Banach spaces. Spaces of continuous linear applications. Dual space. Cauchy-Schwarz inequality. Parallelogram law. Orthogonalization. Hilbert spaces. Fourier series. Bessel's inequality. Parseval equality. Riesz-Fisher theorem. Hahn-Banach theorem. Minkowski function. Separation theorems. General form of linear functionals: functional in spaces of sequences; in Hilbert spaces. Riesz theorem. Weak convergence in a normed space and its dual. Banach-Steinhaus theorem. Compactness in the dual space. Topological vector spaces: Theorem of Kolmogorov. Weak topologies. Banach theorem of the inverse. Symmetric compact applications: Hilbert's Theorem. Fredholm theory.

Research and Events


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Defended Theses

  • Structure, algorithmics and dynamics of endomorphisms for certain classes of groups
      André da Cruz Carvalho (April 2023)
      Pedro V. Silva
  • A point-free study of z-embeddings, more general classes of localic maps, and uniform continuity
      Ana Belén Avilez García (April 2023)
      Jorge Picado
  • Analysis of equations of motion of inextensible strings and networks
      Ayk Telciyan (April 2023)
      Dmitry Vorotnikov
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