The course is aimed at the acquisition of some theoretical and applicative principles of abstract Mathematical Analysis and application to PDEs. 

01. Linear spaces (LS): definition, examples and linear span
01.01 Definition of LS.
01.02 Elementary properties and examples of LS.
01.03 Sublinear spaces, examples and properties.  
01.04 Linear spans. Characterization of the linear span. 

02. Linear spaces: quotient spaces and convex sets
02.01 Definition of a quotient space. The quotient space is a LS.
02.02 Definition of linear maps and isomorphism.
02.03 Linear maps and linear subspaces. 
02.04 Convex sets. 
02.05 Properties of convex sets. Operations.   
02.06 The image (preimage) by a linear map of a convex set is convex. 
02.07 Definition of the convex hull of a set.
02.08 The convex hull of a set S: basic properties. 

03 Normed linear spaces (NLS): definition and basic properties
03.01 Definition of a norm and NLS.
03.02 Distance induced by a norm.
03.03 The topology induced by the norm. Equivalent norms.
03.04 Subspaces of a normed linear spaces are NLS. 
03.05 The quotient space.
03.06 Definition and examples of Banach spaces in PDEs. 

04. Completion of a NLS
04.01 Completion of a NLS: abstract construction.
04.02 Examples of complete NLS. Connection with Partial Differential Equations.
04.03 Examples of noncomplete NLS. 

05. Finite dimensional LS
05.01 Definition of linear independence.  
05.02 Maximal set of linearly independent elements. 
05.03 Norm and zero norm: fundamental inequality.
05.04 Compactness of the unit ball is compact.
05.05 In finite-dimensional linear spaces, all norms are equivalent. 
05.06 Completenees of finite-dimensional NLS. 

06. Non compactness of the unit ball in infinite NLS
06.01 A preliminary lemma.
06.02 In infinite dimensions the unit ball is not compact. 
06.03 Separable spaces.  

Elementary knowledge relating to basic Algebra and Mathematical Analysis is request.  

Knowledge and understanding. At the end of the course the student must have acquired a good knowledge of the mathematical topics covered in the course. He must be able to argue correctly and with language properties on the topics covered in the program. Examples and working methods are shown in the classroom during the lessons and proposed in the exercises.

Applied knowledge and understanding. At the end of the course the student must have acquired a good ability to use the main tools of basic abstract mathematics. He must be able to correctly apply the formulation studied and must be able to solve general mathematical problems similar to those studied. In particular, he must be able to apply the acquired knowledge even in contexts slightly different from those studied, and have the ability to use the acquired knowledge to independently solve problems that may appear new. Examples of such applications are shown in the classroom during the lessons and proposed in the exercises.

Autonomy of judgment. At the end of the course the student must have acquired a good ability to analyze topics and problems in general mathematics, the ability to critically evaluate any proposed solutions, and a correct interpretation of similar topics.

Communication skills. At the end of the course the student must have acquired a good ability to clearly communicate his / her statements and considerations concerning general mathematics problems. The working method is shown in the classroom during the lessons and proposed in the exercises.

Ability to learn. At the end of the course the student must have acquired a good capacity for autonomy in the study of the discipline, in the reading and interpretation of a qualitative phenomenon, in the search for useful information to deepen the knowledge of the topics covered.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

There are no supporting activities.

Teaching

Theoretical lessons and exercises.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory  - Springer, 2007. 

Assessment

The expected objectives are verified through the following mandatory test: an oral interview about the theoretical topics covered in the course.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

Teaching

Theoretical lessons and exercises.

Attendance

Although strongly recommended, course attendance is not mandatory.

Course books

W. Rudin. Principles of Mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

S. Salsa, Partial Differential Equations in Action, From Modelling to Theory  - Springer, 2007. 

Assessment

The expected objectives are verified through the following mandatory test: an oral interview about the theoretical topics covered in the course.

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.