Linear Analysis

Code:

M3003

Acronym:

M3003

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2017/2018 - 2S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	2	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	1	study plan from 2017/18	3	-	6	56	162
L:M	15	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

Study of Linear Analysis in infinite-dimensional spaces.

Learning outcomes and competences

To gain competences of subjects related
to Analysis in different functional  spaces.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Program

1. Linear applications in finite-dimensional spaces: Linear spaces. Linear applications. Structure of a linear application.  Infinite-dimensional spaces. Examples. 
2. Normed  Spaces: Normed Spaces.
Banach spaces. Complete spaces. Compact spaces. 
3. Euclidean spaces: Orthogonal bases. Orthogonalization. Hilbert spaces. Completeness. Fourier series. Inequality of Bessel. Equality of Parseval.
4. Fourier analysis: Fourier trigonometric series.
Point-to-point convergence. Uniform convergence. Fejer's theorem. Weierstrass theorems. Inequality of Bessel. Equality of Parseval. Fourier transform. Laplace transform. Some applications.
5. Linear operators: Linear operators space. Banach-Steinhause's Theorem. Banach's theorem. Compact operators. Fredholm Alternative.
Self-adjoin applications. Hilbert-Schmidt theory. Integral equations.

Mandatory literature

G. Smirnov; Curso de Análise Linear, Porto Editora, 2003

Teaching methods and learning activities

Theoretical and practical classes.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Calculation formula of final grade

Two tests for 10 values each and without normal period examination. The exam is at the time of appeal.