Learning objectives
The course will illustrate the main results from Functional
Analysis, Measure Theory, and from the Lp theory.
Knowledge and understanding: At the end of this course the student should know the essential definitions and results of Functional
Analysis, Measure Theory, and from the Lp theory.
Applying knowledge and understanding: The student should be able to apply the forementioned notions to solve medium level problems, and to understand how they relate to concepts seen in different courses.
Making judgements: The student should be able to evaluate coherence and correctness of the proofs e makes proofs by himself.
Communication skills: The student should be able to communicate in a clear and precise way, suitable for a scientist-to-be in an intermediate stage of his formation.
Lerning skills: The student should be able to link the content of the course with what he learned in his three years bachelor course.
Prerequisites
GENERAL TOPOLOGY, TOPOLOGIA GENERALE (stuff taught in courses Analisi Matematica 1 and Analisi Matematica 2, Geometria 1 eand Geometria 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) Measure Theory: the construction of the Lebesgue measure and and the Lebesgue integral.
11) Lp spaces.
12) Convolutions.
Full programme
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) Measure Theory: the construction of the Lebesgue measure and and the Lebesgue integral.
11) Lp spaces.
12) Convolutions.
DETAILED PROGRAM IN FORMAT DOCX. OR .PDF MAY BE ASKED VIA EMAIL TO ALBERTO.AROSIO@UNIPR.IT - AS WELL AS ANY CLARIFICATION ABOUT THIS COURSE
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, BY MEANS OF SLIDES (=TRANSPARENCIES) AND TRADITIONAL BLACKBOARD. During the lectures the basic results of the functional analysis
will be analyzed and discussed. Many examples and COUNTEREXAMPLES 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
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