# FUNCTIONAL ANALYSIS

Academic Year 2023/2024 - Teacher: Raffaela Giovanna CILIA

## Expected Learning Outcomes

The course aims to provide the basic knowledge of Functional Analysis. Students will  be able to  operate in the framework of topological vector spaces, reflexive spaces,  Hilbert  and   Sobolev spaces.  They learn how the abstract results from Functional Analysis can be applied to solve PDEs.

In particular the course objectives are:

Knowledge and understanding: students will learn the main abstract results in Functional Analysis. They will be able to operate in the framework  of topological vector spaces. The theory of linear operators will be developed and some of the most important classes of spaces will be introduced.

Applying knowledge and understanding: students will be able to apply the mathematical tools learned to solve theoretical  and technical problems.

Making judgements: students will be stimulated  to work on specific topics they have not studied during the class, developing exercises related on the field knowledge with greater independence. Seminars and lectures are scheduled to give students the chance  to share them with the other students.

Communication skills:  students will learn to communicate with clarity and rigour both.

Learning skills: students will be stimulated to examine in depth some mathematical techniques,  arising from the study of the main results,  which may be used to solve other problems.

## Course Structure

The principal concepts and learning outcomes will be structured by planning frontal lectures. Furthermore, to improve the making judgements  homework will be assigned.

## Required Prerequisites

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Knowledge of the main topics of the course Istituzioni di Analisi superiore is useful. Knowledge of Lebesgue measure and integration theory and  the main properties of   Lp  -spaces is recommended.

## Attendance of Lessons

Attendance of the lessons is strongly recommended ( see Regolamento del corso di studi)

## Detailed Course Content

Topological vector spaces. Definition and characterization of topological vector spaces. Filters. Locally convex vector spaces and their characterization. Hausdorff topological vector spaces. Minkowski's functional. Hahn Banach Theorem: analytic and geometric forms. Separation theorems. Extreme points. Krein Milman Theorem.

Linear operators. Continuity. The space of linear continuous operators between normed spaces.  Open mapping Theorem and applications. Closed graph Theorem. Unifom boundedness Theorem. Banach Steinhaus  theorem. Adjoint operator.

Weak topologies. Definition and properties of the weak topology. Closure and weak closure of a convex set. Weak star topology in a dual space. Krein Smulian Theorem. Eberlein Smulian theorem. Banach Alaouglu' s Theorem. Goldstine Theorem.

Reflexive Banach Spaces. Characterizations of reflexive Banach spaces. Separability and weak topologies. Characterization of separable and reflexive Banach spaces. Metrizability of weakly compact spaces. Uniform convex spaces. Milman Pettis Theorem.

Hilbert Spaces. Definitions and elementary properties. Projection onto a closed and convex set. The dual space of a Hilbert space. The Theorems of Stampacchia and Lax-Milgram. Hilbert sums. Orthonormal bases.

Sobolev spaces. Weak derivatives. Definitions and elementary properties of the space  W1,p  . Sobolev inequalities. The Rellich-Kondrakov Theorem. The space  W01,p. Variational formulation of some boundary value problems.

## Textbook Information

1. H. Brezis, Functional Analysis, Sobolev spaces and Partial Differential Equations, Springer

2.  L. V. Kantorovich, G. P. Akilov, Analisi funzionale, Editori Riuniti.

3. R.Megginson An Introduction to Banach space Theory, Springer

4.  H.H. Schaefer, Topological Vector spaces, Springer

## Course Planning

SubjectsText References
1Topological vector spaces and locally convex spaces4
2Linear operators2 or 3
3Weak topologies2 or3
4Reflexive spaces2 or 3
5Hilbert spaces 1
6Sobolev spaces1

## Learning Assessment

### Learning Assessment Procedures

```The final exam consists of an oral test. Final grades will be assigned taking into account the following criteria:Rejected: Basic knowledges have not been acquired. 18-23: Basic knowledges have been acquired. The student has sufficient communications skills and making judgements.24-27: All the  knowledges have been acquired. The student  has good communications skills and making judgements.28-30 cum laude: All the knowledges have been completely acquired. The
student is able to apply  the knowledges to some  new problems. He has
excellent communications skills, learning skills and making judgements.```

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or DSA) of our Department.

### Examples of frequently asked questions and / or exercises

The questions listed below aren't exhaustive, but  just a few examples .

1. Hahn Banach Theorem:  geometric form and applications.

2. Reflexive Banach spaces and some of their characterizations.

3. Example of a topological space that is not locally convex space

4. Hilbert spaces. Projection onto a closed convex set