The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2019) will be open during January 02, 2019 - February 04, 2019
The prize Eng. António de Almeida for the best doctoral thesis defended at FCUP in the years 2015 to 2017 in the areas of Mathematics, Statistics and Computer Science was awarded to Célia Borlido for her thesis "The word problem and some reducibility properties for pseudo- varieties of the form DRH", supervised by Prof. Jorge Almeida within the scope of PIUDM.
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.