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# Linear Analysis

 Code: M3003 Acronym: M3003 Level: 300

Keywords
Classification Keyword
OFICIAL Mathematics

## Instance: 2018/2019 - 2S

 Active? Yes Responsible unit: Department of Mathematics Course/CS Responsible: First Degree 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 0 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 0 study plan from 2017/18 3 - 6 56 162
L:M 10 Official Study Plan 2 - 6 56 162
3
L:Q 0 study plan from 2016/17 3 - 6 56 162

### Teaching Staff - Responsibilities

Teacher Responsibility
Carlos Miguel de Menezes

### Teaching - Hours

 Theoretical and practical : 4,00
Type Teacher Classes Hour
Theoretical and practical Totals 1 4,00
Carlos Miguel de Menezes 4,00
Last updated on 2018-10-22.

Fields changed: Learning outcomes and competences, Palavras Chave, Componentes de Avaliação e Ocupação, Programa

### Teaching language

Suitable for English-speaking students

### Objectives

The student is supposed to acquire during the course proficiency with the main concepts and theorems of Functional Analysis in Banach spaces and Hilbert spaces, with applications to classical function spaces.

### Learning outcomes and competences

1.The student is supposed to acquire during the course proficiency with the main concepts and theorems of Functional Analysis in Banach spaces and Hilbert spaces, with applications to classical function spaces:

a. In Banach spaces: norms, equivalente norms, space of bounded linear maps, dual and bidual space, reflexive spaces, weak topologies,   the so called four pilars of Functional Analysis: theorems of Hahn-Banach, open mapping, closed graph and uniform boundedness principle, and weak topologies. Fixed Point theorems of Banach, Brouwer and Schauder.

b. In Hilbert spaces: orthogonality, Riesz representation theorem, separability, orthonormal families and bases, adjoint operator, projections and idempotents, compact operators, spectral theory of compact normal operators

2. Applications of these results to classical function spaces: sequence spaces, continuous functions spaces,  and integrable functions, and, if time allows, application of spectral theory of compact self adjoint operators to Sturm-Liouville systems. Several examples of rlevant operators and functional spaces will be the subject of problems/exercises to be solved by the student in practice lectures or autonomus at home.

Presencial

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

Real Analysis/Calculus  in one and several variables, Complex Analysis, basic notions on metric spaces aquired in real analysis courses, Linear Algebra in finite dimensions.

Have attended an introdcutory course in topology/metric spaces will be a great help.

### Program

I. Real and Complex  Normed spaces and  Banach spaces:

1. General concepts: norm, associated distance, open and closed balls.   Sequences and series in a  normed space. Cauchy sequences and fast Cauchy sequences. Banach spaces. Bounded linear maps.  The normed space of  bounded linear maps between two normed spaces. Banach fixed point theorem.

2. Equivalent norms. All norms in a finite dimensional vector space are equivalent. locally compact normed spaces are the finite dimensional normed spaces.  Sums and products of normed spaces, norm in the quotient of a normed space by a closed subspace.

3. Examples of function spaces with the norm of uniform convergence:
bounded functions, bounded continuous functions,  continuous functions with compact support,  continuous function sthat vanish at infinity. Banach Spaces of $C^k$ functions. Study of completeness of the above spaces.
Sequence spaces related to previous spaces as particular cases.

4. Dini theorem for monotone  sequence of continuous functions defined in a compact space and with real values. Stone-Weierstrass theorem.

5.  Equicontinuity and Ascoli-Arzéla theorem.

6. Given a  bounded continuous map defined in a dense subspace of a Banach space X with values in another Banach space Y, extension of that map to the Banach space X.  Completion of a  normed space.

7. Baire category theorem. Open mapping theorem. Banach isomorphism theorem.

8.  Closed graph theorem. Uniform boundedness principle. Banach-Steinhaus theorem.

9. Brouwer and Schauder fixed point theorems.

10. Topological dual and  bidual  of a normed space. Hahn-Banach theorem in analytic and geometric form. Reflexive Banach spaces. Transpose (or adjoint) of a bounded linear map. Closed range theorem.

11. Weak and weak^*-convergence in a  Banach space. Banach-Alaoglu theorem.

12. Young, Holder and Minkowski inequalities (discrete and contínuous).
Space of Lp-Riemnan integrable functions in compacts of R^d. Proof that for  these normed spaces are not complete (1<=p< \infty)

13. Lebesgue integral in euclidean spaces as  extension of Riemann integral to the completion in L^1 norm of the space of continous functions with compact support.

II. Espaços de Hilbert:

1. General concepts: Sesquilinear forms, inner products.
Polarization identity,  Cauchy-Schwarz inequality, norm associated to the inner product, paralelogram law, orthogonal vectors, Pitagoras theorem.
Prehilbertian and Hilbert spaces.
Orthogonality and biorthogonality of vector subspaces.

2. Orthogonal  sum of Hilbert spaces. Orthogonal projection
on a convex closed set and associated orhogonal decomposition of the Hilbert space. Riez representation theorem of the topologiacl dual of a Hilbert space. Completion of a pre-Hilbert space. Weak topology  in a Hilbert space.

3. Total family, orthonormal family, and orthonormal basis of a
Hilbert space. Extension of orthonormal family to a orthonormal basis.
Dimension of the Hilbert space. Separable Hilbert spaces.
Gram-Schmidt orthonormalization of total famíly. Bessel inequality, Parseval identity, Fourier series in Hilbert spaces. Isometric Hilbert spaces.

4. Adjoint of a bounded inear operator between Hilbert spaces and its main proprerties. Self adjoint and normal operators. Characterization of bounded invertible linear operators in terms of the adjoint.

5.  Projections, idempotents, unitary operators, partial isometries, invariant and resctriction subspaces.

6. Compact operators in Hilbert spaces.

7. Spectral theory of compact  self adjoint and normal operators.

### Mandatory literature

Carlos Menezes; Apontamentos de Análise Linear-2018-2019

### Complementary Bibliography

Kreyszig; Introductory functional analysis with applications
Pedersen, Gert; Analysis Now, Springer

### Teaching methods and learning activities

Presencial lectures on the contents of the syllabus given by the teacher.

The students will work in classes and at home on exercises and problems proposed  by the teacher.

### keywords

Physical sciences > Mathematics > Mathematical analysis > Functional analysis

### Evaluation Type

Evaluation with final exam

### Assessment Components

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

### Amount of time allocated to each course unit

designation Time (hours)
Estudo autónomo 106,00
Frequência das aulas 56,00
Total: 162,00

### Eligibility for exams

The attending of classes is not mandatory.

### Calculation formula of final grade

The final classification will be the classification of the final exam.

### Special assessment (TE, DA, ...)

Any type or special examination can be from one the following types: exclusively by an oral examination,  only  a written exam, one oral examination and a written exam.

The decision of which of the above types is each special examination  is exclusively the responsability of the teacher  assigned to the curricular unit.