Mathematics | University of Parma

Professor
Academic discipline: Analisi matematica (MAT/05)
Field: Formazione teorica
Type of training activity: Characterising

56 hours
of face-to-face activities

6 credits

hub: PARMA

course unit
in - - -

lectures timetable unavailable

exam calls

Learning objectives

The course it is aimed at illustrating the main results from Functional
analysis and from the Lp theory.

Prerequisites

Analysis 1 and Analysis 2

Course unit content

1) Normed spaces and Banach spaces
2) Operators on normed spaces
3) Hahn-Banach theorem and its consequences.
4) Banach-Steinhaus theorem and its consequences
5) Open mapping theorem and its consequences
6) Weak topologies in Banach spaces
7) Reflexive spaces
8) Hilbert spaces: main properties, projections, orthonormal systems.
9) Application: Fourier Series.
10) Introduction to Measure Theory: the construction of the Lebesgue measure and and the Lebesgue integral.
11) Lp spaces.
12) Convolutions.

Full programme

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Bibliography

1) H. Brezis. Functional analysis, Sobolev spaces and partiare differential
equations, Springer Verlag 2011

2) W. Rudin. Real and complex Analysis. McGraw-Hill Book Co., New York, 1987

Teaching methods

Lectures. During the lectures the basic results of the functional analysis
will be analyzed and discussed. Many examples will be provided to show
how and where the
abstract results can be applied to make the students understand better
the relevance of what they are studying.

Assessment methods and criteria

The exam consists of two parts: a written part and an oral part. The exam
is aimed at evaluating the knowledge of the abstract results seen during
the course, their proofs and the skills of the students in using such
results.

Other information