Introduction to Functional Analysis. Fall 2019
Syllabus
- Lecture 1 (06.09.2019).
- Normed spaces. Examples: norms on finite-dimensional spaces,
𝓁^{p}, c_{0}, 𝓁^{∞}(X), C_{b}(X),
C^{n}[a,b], L^{p}(X,μ). Bounded linear operators.
Characterizations of bounded linear operators. Boundedness = continuity.
The domination relation and the equivalence of norms on a vector space.
- Lecture 2 (13.09.2019).
- Any two norms on a finite-dimensional vector space
are equivalent. The norm of a bounded linear operator. Topologically injective operators,
isometries, topological and isometric isomorphisms. Topologically injective = bounded below.
Examples of bounded linear operators:
multiplication operators, shift operators, integral operators.
- Lecture 3 (20.09.2019).
- Banach spaces. Examples: finite-dimensional
spaces, 𝓁^{∞}(X), C_{b}(X), 𝓁^{p},
L^{p}(X,μ). The space ℬ(X,Y) of bounded linear operators is complete
whenever Y is complete. The "extension by continuity" theorem.
Completions of normed spaces: existence, universal property, uniqueness, functoriality.
- Lecture 4 (27.09.2019).
- Open operators and coisometries.
Characterizations of open operators. Quotients of normed spaces. The universal property
of quotients. Corollaries. The completeness of quotients. Inner producr spaces.
The Cauchy-Bunyakowski-Schwarz inequality. The norm generated by an inner product.
Hilbert spaces. Examples.
- Lecture 5 (04.10.2019).
- Unitary isomorphisms of inner product spaces.
Orthogonal complements and their basic properties. Orthogonal projections: equivalent definitions.
The existence of orthogonal projections onto a closed subspace of a Hilbert space.
The orthogonal complement theorem. Orthonormal families. Examples. Fourier coefficients
and their geometric properties. Bessel's inequality. Fourier series and their elementary properties
(uniqueness, Parseval's identity).
- Lecture 6 (11.10.2019).
- Orthonormal bases, total orthonormal families,
maximal orthonormal families. Relations between these notions.
Orthogonalization. The existence of an orthonormal basis in a separable inner
product space. Examples of orthonormal bases. Classification of separable Hilbert spaces.
The Riesz-Fischer theorem.
The Hahn-Banach theorem for real vector spaces and for linear functionals dominated by
sublinear functionals.
- Lecture 7 (18.10.2019).
- The Hahn-Banach theorem for
seminormed spaces. Corollaries (for normed spaces): extension of bounded linear functionals,
a "dual" formula for the norm, linear functionals separate the points,
linear functionals separate the points and vector subspaces.
Barrels in normed spaces. The barrel lemma for Banach spaces.
The Uniform Boundedness Principle (the Banach-Steinhaus theorem).
- Lecture 8 (25.10.2019).
- The Open Mapping Theorem, the Inverse Mapping Theorem,
the Closed Graph Theorem. Dual spaces and dual operators. Basic properties of dual operators.
The Riesz Representation Theorem on linear functionals on Hilbert spaces.
The dual of 𝓁^{p}.
- Lecture 9 (08.11.2019).
- Similar and isometrically equivalent operators.
The duals of diagonal and shift operators on 𝓁^{p}. Complex measures.
The variation of a complex measure. σ-additivity implies bounded variation.
The integral of a bounded measurable function w.r.t. a complex measure.
The dual of the space B(X) of bounded measurable functions (the Hildebrandt-Kantorovich theorem).
Complex Radon measures on compact topological spaces. The dual of C(X) (the Riesz-Markov-Kakutani theorem,
without proof). A special case: the dual of C[a,b] in terms of functions of bounded variation
(the Riesz theorem, without proof).
- Lecture 10 (15.11.2019).
- The canonical embedding of a normed space into the bidual.
Reflexive Banach spaces. Examples. Annihilators and preannihilators, their basic properties.
The dual of a subspace and of a quotient. Relations between kernels and images of operators and of their duals.
A duality between injective operators and operators with dense image. A duality between topologically injective
and surjective operators on Banach spaces. A duality between isometries and coisometries
on Banach spaces.
- Lecture 11 (22.11.2019).
- The spectrum of an algebra element. Examples.
Algebraic properties of the spectrum (the behavior of the spectrum under homomorphisms, the polynomial spectral
mapping theorem, the spectrum of the inverse element). Banach algberas. Examples. Properties of the multiplicative
group of a Banach algebra. The automatic continuity of characters (i.e., of C-valued homomorphisms)
of a Banach algebra.
- Lecture 12 (29.11.2019).
- The compactness of the spectrum of a Banach algebra element.
The resolvent function and its properties. The nonemptiness of the spectrum. The Gelfand-Mazur theorem.
The spectral radius of a Banach algebra element. The point spectrum, the continuous spectrum, and the residual
spectrum of a bounded linear operator. An example: calculating the parts of the spectrum for the diagonal
operator.
- Lecture 13 (06.12.2019).
- Spectra of similar operators. The spectrum of the dual operator.
Relations between the parts of the spectrum of an operator and of the dual operator.
An example: the parts of the spectrum of the shift operators on 𝓁^{p}.
Totally bounded metric spaces. Examples and counterexamples.
The compactness criterion of a metric space (compactness ⇔ sequential compactness ⇔ countable compactness
⇔ limit point compactness ⇔ total boundedness + completeness) (statement only).
Corollaries. Uniform continuity and equicontinuity. The Cantor theorem and the Arzela-Ascoli theorem
(statements only). The Riesz lemma on an ε-perpendicular. The noncompactness of the sphere
in an infinite-dimensional normed space.
- Lecture 14 (13.12.2019).
- Compact operators: definition, simplest examples and counterexamples.
Properties of the set of compact operators. Schauder's theorem on the compactness of the dual operator.
Approximation of Hilbert-space-valued compact operators by finite rank operators. Remarks on the approximation property
and Schauder bases. The compactness criterion for a diagonal operator on 𝓁^{p} or on c_{0}.
The adjoint of an operator between Hilbert spaces. Basic properties of the operation of taking the adjoint.
The C*-property.
- Lecture 15 (20.12.2019).
- Selfadjoint operators.
The spectrum of a selfadjoint operator is real. The orthogonality of the eigenspaces of a selfadjoint operator.
The spectral radius of a selfadjoint operator equals the norm.
A relation between invariant subspaces on an operator and of its adjoint.
The quadratic form of a selfadjoint operator. The norm of a selfadjoint operator in terms of its
quadratic form. The norm of a compact selfadjoint operator in terms of eigenvalues.
The Hilbert–Schmidt theorem.
- 20.12.2019. Individual problem discussion
- 13.12.2019. Traditional seminar
- 06.12.2019. Individual problem discussion
- 29.11.2019. Traditional seminar
- 22.11.2019. Individual problem discussion
- 15.11.2019. Traditional seminar
- 08.11.2019. Individual problem discussion
- 25.10.2019. Individual problem discussion
- 18.10.2019. Traditional seminar
- 11.10.2019. Individual problem discussion
- 04.10.2019. Traditional seminar
- 27.09.2019. Individual problem discussion
- 20.09.2019. Traditional seminar
- 13.09.2019. Individual problem discussion (прием задач)
- 06.09.2019. Traditional seminar (problem discussion at the blackboard)
You are allowed to tell your solutions to an instructor
after the deadline as well, but you earn half the points for that.
Exercises marked by "-B" are optional.
If you solve such exercises, you will earn bonus points.
- Exercise sheet 1. Normed spaces. Deadline 11.10.2019.
- Exercise sheet 2. Linear operators. Banach spaces. Deadline 08.11.2019.
- Exercise sheet 3. Hilbert spaces. The three basic principles
of Functional Analysis. Deadline 06.12.2019.
- Exercise sheet 4. Duality for normed spaces. Spectra of linear operators.
Deadline 23.12.2019.
The followng two lists are not exercise sheets, but are just selections of exercises some of which
were discussed at the blackboard during traditional seminars.
They can also be helpful for preparing to the exam.