Theory of Distributions


By the end of this course, the student should have acquired knowledge, skills and competences in order to:

-Use fluently the elementary properties, results and procedures concerning distributions, in view of applications to

differential equations;

-Understand the topology of the distribution''''s space and apply it to the different notions of convergence of a sequence of functions.

-Differentiate distributions and establish the relationship between this type of differentiation and differentiation in the

classical sense.

-To be familiar with the notion of differential operator and fundamental solution.

-Apply the Fourier Transform to tempered distributions and to know its main properties.


General characterization





Responsible teacher

José Maria Nunes de Almeida Gonçalves Gomes


Weekly - Available soon

Total - 56

Teaching language



Knowledge on classical differential and integral calculus at a graduate level. Basic notions of toplogy and Functional  Analysis. Knowledge of the Lebesgue integral and of fundamental notions on Measure Theory.  


1.F. Friedlander and M. Joshi, Introduction to the Theory of distributions;

2.A. Kolmogorov and V. Fomin, Introductory real analysis;

3.W. Rudin, Functional Analysis;

4.R. Strichartz, A Guide to Distribution Theory and Fourier Transforms;


5.L. Evans and M. Gariepy; Measure Theory and Fine Properties of Functions;

5.L. Schwartz, Théorie des Distributions;

6.L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques.

7.Yosida, Functional Analysis.

Teaching method

Theorical/Problem solving sessions complemented by discussion sessions.

Evaluation method

Final exam evaluation.

Subject matter

1. Historical Context.

Differentiation of nonregular functions.

2. Preliminaries.

Frechet''''s Spaces; Spaces C^k and C^{\infty}, Inductive topological limits and test functions.

3. Space of Distributions.

Linear Continuous forms; Locally Integrable Functions.

4. Convergence of sequences of distributions; Weak and Weak * topology; Caracterisation of the Convergence of

Distributions and fundamental properties.

5. Differentiation of distributions.

Differential operators; Fundamental Solutions; Convolution of distributions; General properties of convolution;

6. Tempered Distributions.

Smooth rapid descreasing functions; L^P spaces as tempered distributions; Properties of the Fourier Transform.

7. Application: Sobolev Spaces.


Programs where the course is taught: