ID:
500659
Duration (hours):
78
CFU:
9
Year:
2025
Overview
Date/time interval
Primo Semestre (22/09/2025 - 09/01/2026)
Syllabus
Course Objectives
The aim of the course is to guide the student through the principles and tools of abstract Functional Analysis and (some of) its important applications. By the end of the course, the student will have acquired the knowledge and skills necessary to begin formulating and studying Mathematical Analysis problems in infinite-dimensional settings.
Course Prerequisites
-Differential calculus -Measure theory and Lebesgue's integration -Linear algebra and general topology
Teaching Methods
Lectures and tutorials
Assessment Methods
The exam consists of a written test and an oral examination. The written test includes the solution of several exercises and, possibly, answers to theoretical questions. Its purpose is to assess whether the student has understood and mastered the concepts, tools, and techniques developed during the course. The oral examination, which can only be taken if the written test is passed, complements the written part and provides a deeper assessment of the student’s understanding of the topics covered in the course. The final grade will be based on the overall breadth and depth of learning, as well as on the clarity of presentation and the skills demonstrated in problem-solving. It will result from a combination—not necessarily a simple arithmetic average—of the evaluations of the written and oral components.
Texts
-H. Brézis: Analisi Funzionale, Liguori Editore. -G. Gilardi: Analisi Funzionale. Mc Graw Hill, 2014 H. Brézis: Functional analysis, Sobolev spaces and partial differential equations. Springer, 2011.
Contents
Review of norms and inner products. Normed spaces. Linear and continuous operators. Topological dual. Banach spaces. The Hahn–Banach Theorem: analytic and geometric forms, and their consequences. Baire’s Lemma. Banach–Steinhaus Theorem. Open Mapping Theorem, Closed Graph Theorem, and their consequences. Weak* topology, weak topology, and their properties. Banach–Alaoglu Theorem. Reflexive spaces. Separable spaces. L^p spaces. Reflexivity and separability in L^p spaces. Riesz Representation Theorem. Approximation by convolution. Ascoli–Arzelà Theorem. Fréchet–Kolmogorov Theorem. Hilbert spaces. Projection onto a closed convex set. Riesz Representation Theorem for the dual. Lax–Milgram Theorem. Complete orthonormal systems. Compact operators. Adjoint of a bounded operator. Fredholm Alternative Theorem. Spectrum of a compact operator. Spectral decomposition of a compact self-adjoint operator. Sobolev spaces in one dimension. Introduction to Sobolev spaces in dimension N. Applications to linear elliptic partial differential equations.
Course Language
Italian
Degrees
Degrees
PHYSICAL SCIENCES
Master’s Degree
2 years
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