Functional Analysis

Code:

M503

Acronym:

M503

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 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	8	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

The main objective is to make a general introduction to Functional Analysis, in order to provide the student with the tools to study and understand further issues in different areas such as Analysis, Differential Equations, Probability Theory or Ergodic Theory.

Learning outcomes and competences

Familiarity with results and concepts about functionals and operators in Banach and Hilbert spaces.

Working method

Presencial

Program

1. Hahn-Banach Theorem
   
   - Analytic form. Extensions of lineare functionals.
   - Geometric form. Separation of convex sets.
   - Bidual. Orthogonality relations.
   
   


2. Operators in Banach spaces
   
   - Uniform Boundedness Principle.
   - Open MappingTheorem and Closed Graph Theorem.
   - Complementary spaces. Right and left inverses.
   - Unbounded operators. Densely defined operators. Adjoint Operator. Closed operators.
   - Characterization of operators with closed range.
   - Characterization of surjective operators.
   
   


3. Weak topologies and reflexivity
   
   - Elementary properties of the weak topology.
   - Weak topology, convex sets and linear operators.
   - Weak* topology.
   - Reflexive spaces. Relation to weak topologies.
   - Separable spaces. Relation to reflexivity and weak topologies.
   - Uniformly convex spaces.
   
   


4. L^p spaces
   
   - General properties on measure and integration.
   - Elementary properties on L^p spaces.
   - Reflexivity. Separability. Duality.
   - Convolution and regularization.
   
   


5. Hilbert spaces
   
   - Definition and elementary properties.
   - Projection on a convex closed set.
   - Dual space.
   - Stampacchia and Lax-Milgram Theorems.
   - Hilbert sums. Orthonormal bases.
   
   


6. Compact operators
   
   - Definition and elementary properties.
   - Adjoint operator.
   - Riesz-Fredholm Theory.
   - Spectrum.
   - Spectral decomposition of self-adjoint compact operators.

Mandatory literature

Brezis, Haim; Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. ISBN: 978-0-387-70914-7
Haim Brézis; Analyse fonctionnelle. ISBN: 2-225-77198-7

Complementary Bibliography

Elements of Theory of Functions and Functional Analysis. ; A.N. Kolmogorov, S.V. Fomin
Walter Rudin; Functional analysis. ISBN: 0-07-054225-2
John B. Conway; A course in functional analysis. ISBN: 0-387-96042-2

Teaching methods and learning activities

Each lecture lasts for 2 hours. Whenever possible, the last half hour of each one will be spent with exercises. Pertinent list of exercises will be provided to students.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	70,00
Trabalho escrito	30,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	30,00
Frequência das aulas	58,00
Trabalho escrito	5,00
Total:	93,00

Eligibility for exams

Does not apply.

Calculation formula of final grade

Evaluation will be held by a final exam with a weight 14 out of 20 and homework assignements with the remaining 6 points.