Functional Analysis

Code:

M503

Acronym:

M503

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	5	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

To know elements of theory of metric, normed and Banach spaces, various type of convergences and property of bounded linear operators. To apply methods of functional analysis to solve integral equations of Fredholm’s and
convolution type.

Learning outcomes and competences

Knowledge of Functional Analysis.

Working method

Presencial

Program

Definition and examples of: metric, complete, normed, Banach, Euclidean, Hilbert and topological vector spaces.The space of continuous linear maps; dual space; adjoint map.Results: Banach fixed point theorems; the nested balls theorem; theorem of Baire; theorem of Arzelà-Ascoli; Cauchy-Schwarz inequality; parallelogram law; orthogonalization; Bessel’s inequality; Parseval’s identity; the theorem ofRiesz-Fisher; Fourier series; Riemann’s lemma; pointwise convergence; continuous maps whose Fourier series diverge in a point; convexity; Zorn’s lemma; theorem of Hahn-Banach; weak convergence; the theorems of Banach-Steinhausand Mazur; inverse function theorem; compact symmetric maps; theorem of Hibert.

Mandatory literature

Elements of Theory of Functions and Functional Analysis. ; A.N. Kolmogorov, S.V. Fomin

Teaching methods and learning activities

Final exam.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Calculation formula of final grade

Till 20 points of scores.