ADVANCED ANALYSIS A
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
- Contacts/Info
The content of Mathematical Analysis 1-3, Linear Algebra and Geometry, Geometry 1
- Homework exercises which will verify the acquisition of an operational knowledge of the subject and the ability to apply the techniques illustrated in class to produce indpendently proofs of statements similar to those seen in in the lectures and to express themselves in rigorous mathematical langauge.
- Final oral exam devoted to the discussion of the homework exercises, and to the proof of one or two theorems seen in class. This part will assess the acquisition of an in-depth knowledge of the topics presented in class and the students' ability to express themselves in a rigorous mathematical language and to recognize the validity of, even subtle, mathematical reasonings.
The aim of the course aims is to deepen the study of modern analysis begun in the previous courses.
Students will acquire a working knowledge of the methods of advanced analysis. They will know statements and proofs of the main theorems, and will be able to solve exercises, even of theoretical nature, on the topics treated in the course. They will have learned a number of techniques of proof which they will be able to use to recognize the validity of sophisticaded mathematical reasoning and to prove results related to those described in class. Finally students will be able to express themselves in a rigorous mathematical language.
Theory of operators in HIlbert spaces: Introduction to Banach spaces. Linear operators in normed spaces. Finite dimensional spaces. Linear functionals and the Hahn-Banach theorem. Baire's theorem and consequences: Banach-Steinhaus, open mapping, and closed graph theorems. Strong and weak convergence.
Hilbert spaces. Projection theorem, orthonormal bases. Linear operators and their adjoints. Projection, isometric and unitary operators.
Spazi di Hilbert: Teorema della proiezione, basi ortonormali. Finite rank and compact operators. Hilbert-Schmidt operators. Closed and closable operators. Symmetric self adjoint and normal operators. Self-adjoint extensions of symmetric operators.
Basic notions of spectral theory. Spectral theory of compact operators. The spectral theorem for self-adjoint and normal operators.
One parameter groups of unitariy operarors.
Schroedinger operators and applications to PDE's.
Introduction to Banach spaces. Linear operators in normed spaces. Finite dimensional spaces. Linear functionals and the Hahn-Banach theorem. Baire's theorem and consequences: Banach-Steinhaus, open mapping, and closed graph theorems. Strong and weak convergence.
Hilbert spaces. Projection theorem, orthonormal bases. Linear operators and their adjoints. Projection, isometric and unitary operators.
Spazi di Hilbert: Teorema della proiezione, basi ortonormali. Finite rank and compact operators. Hilbert-Schmidt operators. Closed and closable operators. Symmetric self adjoint and normal operators. Self-adjoint extensions of symmetric operators.
Basic notions of spectral theory. Spectral theory of compact operators. The spectral theorem for self-adjoint and normal operators.
One parameter groups of unitariy operarors.
Schroedinger operators and applications to PDE's.
H. Lal Vasudeva, Elements of Hilbert Spaces and Operator Theory, Springer.
M. Reed B. Simon, Methods of Modern Mathematical Physics, Vol. 1 - Functional Analysis, Academic Press
M. Reed B. Simon, Methods of Modern Mathematical Physics, Vol 2 - Fourier Analysis, Self-Adjointness, Academic Press
W. Rudin, Real and Complex Analysis, Mc Graw Hill.
G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrodinger Operators, GSM AMS
J. Weidmann, Linear operators in Hilbert spaces, Springer.
Frontal lectures: 64 hours
Office hours: by appointment (email the instructor)
Professors
Borrowers
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Degree course in: MATHEMATICS
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Degree course in: MATHEMATICS