- Docente: Giovanna Citti
- Credits: 6
- SSD: MATH-03/A
- Language: Italian
- Teaching Mode: In-person learning (entirely or partially)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 6061)
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from Sep 23, 2026 to Dec 17, 2026
Learning outcomes
By the end of the course, students will be familiar with a number of advanced and modern tools in mathematical analysis. They will understand the foundations of the theory of Hilbert and Banach spaces, (L^p) spaces, and Fourier transforms and series. They will be able to apply this knowledge to address and solve problems arising in pure and applied sciences. They will possess the knowledge and skills in mathematical analysis required for admission to second-cycle degree programmes.
Course contents
Hilbert and Banach Spaces
Definition of Banach and Hilbert spaces
Bases in Hilbert spaces: every separable Hilbert space has a countable basis
Examples and exercises
Compactness Theorems
The Arzelà–Ascoli theorem
L p L^p Spaces
Completeness of L p L^p, 1 ≤ p < ∞ 1 \leq p < \infty
Convolution and mollifiers
The space L ∞ L^\infty
Examples and exercises
Linear Operators between Hilbert and Banach Spaces
Linear operators and the definition of compact operators
The representation theorem for linear functionals on a Hilbert space
The Hahn–Banach theorem
Examples and exercises
Weak Convergence in Hilbert and Banach Spaces
The Baire category theorem and the uniform boundedness principle
Weak convergence
Examples and exercises
The Fourier Transform
Definition and main properties in L 1 L^1
Definition in L 2 L^2
Examples and exercises
Office hours
See the website of Giovanna Citti