Topic outline
Analysis AA 2024/2025
Teachers: Prof. Elena Cordero and Prof. Joerg Seiler
elena.cordero@unito.it, joerg.seiler@unito.it
Teaching Assistant: Stefano Baranzini, stefano.baranzini@unito.it
Teaching Activities: Tuesdays 4:30 pm-8 pm, Room 10 (starting from October, 1st)
Examination Rules
EXAMINATION RULES
The examination is composed by a written and an oral test. The written test consists in open questions and exercises on the topics treated in class and has a duration of 180 minutes (3 hours). The mark will be expressed in thirtieth; the single points (30 in total) will be distributed to the questions and exercises on the basis of their importance and length; the final score will be given by summing up the partial scores of each question and exercise. The oral examination is scheduled after the written test and can be given only after having passed the written test with a mark of 18 or better. The oral examination consists of questions on the written test and on the topics treated in class and listed in the examination programme.
The oral examination is not mandatory. Students can decide whether to accept the mark of the written test as final score or to take the oral test. This decision must be communicated to the teachers soon after the written test's results.
PDF Lectures Prof.ssa Cordero ay 21-22
Preliminaries
Introduction to the course. Basic concepts from linear algebra. Linear spaces and examples. Linear subspaces and examples. The linear space of functions from a set to a linear space. Linear operators. The kernel and the image of a linear operator.
Metric spaces, definition and main examples. Sequences and subsequences. Convergence of sequences. Open and closed sets. The closure of a set and its characterization. Dense sets. Continuity of functions between metric spaces. Characterization of compactness by means of sequences. The Heine-Cantor Theorem.
Weierstrass Approximation Theorem. Main tools for the proof. Approximate Identity. The convolution. Approximate identity and convolution with functions that are continuous and bounded.
Normed Spaces
Normed-spaces. Basic Examples
Finite-dimensional normed spaces.
Riesz' Lemma
If X is an infinite-dimensional normed space then neither the unit disc nor the unit circle is compact.
Banach spaces
Inner Product Spaces
Inner Products
Orthogonality
Any finite-dimensional vector space admits an orthonormal basis
The generalization of Pythagoras' Theorem for a k-dimensional inner product space
Orthogonal Complements, basic properties. Characterization of the orthogonal complement for linear subspaces.
The Projection Theorem
The Orthogonal Decomposition Theorem
Orthonormal sequences.
Bessel's Inequality
Orthonormal Bases in Infinite Dimensions. Equivalent definitions. Parseval Theorem
Fourier Series
Linear Operators
Characterization of continuous linear operators
The space B(X,Y) of continuous linear operaators between the normed space X and Y. Equivalent norms.
Basic Examples.
If T is in B(X,Y), then Ker T, the graph of T: G(T) are closed.
A matrix A is a linear, bounded operator.
The Density Principle.
Definition of isometry, examples.
Definition of isometric isomorphism.
Theorem of Riesz-Fischer.
X normed space, Y Banach space, then B(X,Y) is a Banach space.
The dual space X' of a normed space X.
B(X) is an algebra with identity.
The Inverse of a linear bounded operator. Example.
Definition of isomorphic normed spaces.
The invertibility of I-T for T in B(X), X Banach space, with ||T||<1. The Neumann series. Example.
The Open Mapping Theorem.
The Banach Isomorphism Theorem.
The Closed Graph Theorem.
Characterization of invertible operators. Examples
The Uniform Boundedness Principle
Linear operators on Hilbert spaces
Definition and properties of the adjoint of a linear continuous operator between Hilbert spaces.
Examples and exercises of computation of the adjoint of linear bounded operators on Hilbert spaces.
Normal, unitary and self-adjoint operators. Positive operators and their square roots. Projections.
Spectrum and resolvent of a bounded linear operator between Hilbert spaces. Properties of the spectrum. Point, continuous and residual spectrum.Examples and exercises of computation of the spectrum of linear bounded operators on Hilbert spaces.
Dual spaces
Definition of the dual of a normed space
The dual space for a finite dimensional vector space
The Riesz-Frechet Theorem
Consequences: the dual of a Hilbert space
Examples of dual spaces. The case of l^p, L^p(X), for 1\leq p<infty
The Hahn-Banach Theorem in normed spaces
Consequences of the Hahn-Banach Theorem: the norm of a vector x in X expressed by means of the functionals f\in X'
Definition of the bidual X'' of a normed space X
The canonical isometry J, from X to X''
Definition of reflexive space. Examples.
Projections and complementary subspaces
Weak converge in X, Banach space. Examples. Main properties.
Characterization of weak convergence
Convergence in the dual X': the norme convergence, the weak convergence, the weak-* convergence. Relations among different types of convergence.
Main properties of weak-* convergence.
The Banach-Alaoglu Theorem. Corollary.
Compact operators
Definition and properties of compact operators between Banach spaces.
Spectral theory for compact, self-adjoint operators between Hilbert spaces.
Applications: solution of integral equations; solution of boundary value problems for ordinary differential equations and of the corresponding eigenvalue problem.
Fixed point theorems
Strict contractions in a metric space.
The Banach fixed point theorem.
Application: Cauchy problem for first order ODEs.
Distributions and the Fourier transform
The space D(A) of test functions, A open subset of R^n.
The space D'(A) of distributions, A open subset of R^n.
Examples: The Dirac delta distribution, regular distributions, principal value of 1/x.
The support of a distribution. Distributions with compact support.
Differentiation of distributions, multiplication of distributions by smooth functions.
Convolution of functions and distributions.
Distributions and partial differential equations. Fundamental solution.
The space of rapidly decreasing functions S(R^n).
The Fourier transform of L^1-functions.
The Fourier transform and its inverse for rapidly decreasing functions.
The Fourier transform of L^2-functions.
Parseval's formula and Plancherel's theorem.
The space of tempered distributions S'(R^n).
Fourier transform of tempered distributions.
Fourier transform and partial differential equations.
The Heisenberg uncertainty principal