Introduction to Functional Analysis. Fall 2020

Alexei Yu. Pirkovskii (lectures), Ivan S. Shilin and Vladislav V. Balakirev (exercises)

Syllabus

Lectures

Lecture 1 (17.09.2020) (video)
Normed spaces. Examples: norms on finite-dimensional spaces, 𝓁p, c0, 𝓁∞(X), Cb(X), Cn[a,b], Lp(X,μ). Bounded linear operators. Characterizations of bounded linear operators. Boundedness = continuity. The domination relation and the equivalence of norms on a vector space. Any two norms on a finite-dimensional vector space are equivalent.
Lecture 2 (24.09.2020) (video)
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. Banach spaces. Examples: finite-dimensional spaces, 𝓁∞(X), Cb(X).
Lecture 3 (01.10.2020) (video)
The completeness of 𝓁p. 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 (08.10.2020) (video)
Open operators and coisometries. Characterizations of open operators. Quotients of normed spaces. The universal property of quotients. Corollaries. The completeness of quotients.
Lecture 5 (15.10.2020) (video)
Inner product spaces. The Cauchy-Bunyakowski-Schwarz inequality. The norm generated by an inner product. Hilbert spaces. Examples. 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.
Lecture 6 (22.10.2020) (video)
Orthonormal families. Examples. Fourier coefficients and their geometric properties. Bessel's inequality. Fourier series and their elementary properties (uniqueness, Parseval's identity). 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.
Lecture 7 (29.10.2020) (video)
The Hahn-Banach theorem. Corollaries: extension of bounded linear functionals, a "dual" formula for the norm, linear functionals separate the points, linear functionals separate the points and vector subspaces. Dual operators. Basic properties of dual operators. The Riesz Representation Theorem for linear functionals on a Hilbert space.
Lecture 8 (12.11.2020) (video)
The dual of 𝓁p. Similar and isometrically equivalent operators. The duals of diagonal and shift operators on 𝓁p. Complex measures. The variation of a complex measure. The integral of a bounded measurable function w.r.t. a complex measure. The dual of the space B(X) of bounded measurable functions. Complex Radon measures on compact topological spaces. The dual of C(X) (the Riesz-Markov-Kakutani theorem, without proof). The canonical embedding of a normed space into the bidual. Reflexive Banach spaces. Examples.
Lecture 9 (19.11.2020) (video)
Annihilators and preannihilators, their basic properties. The double annihilator theorem. Relations between kernels and images of operators and of their duals. A duality between injective operators and operators with dense image. Barrels in normed spaces. The barrel lemma for Banach spaces. The Uniform Boundedness Principle (the Banach-Steinhaus theorem). Corollaries.
Lecture 10 (26.11.2020) (video)
The Open Mapping Theorem, the Inverse Mapping Theorem, the Closed Graph Theorem. The spectrum of an algebra element. Examples. The behavior of the spectrum under homomorphisms. Spectrally invariant subalgebras.
Lecture 11 (03.12.2020) (video)
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. The compactness of the spectrum. The resolvent function and its properties. The nonemptiness of the spectrum. The Gelfand-Mazur theorem.
Lecture 12 (10.12.2020) (video)
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. 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 for the shift operators on 𝓁p.
Lecture 13 (17.12.2020) (video)
The Riesz lemma on an ε-perpendicular. The noncompactness of the sphere in an infinite-dimensional normed space. A survey of compact operators: definition, basic properties, examples, approximation of compact operators on a Hilbert space by finite rank operators, remarks on the approximation property and Schauder bases, the Riesz–Schauder theory of operators "1+compact", the abstract Fredholm theorems and the Fredholm alternative, the adjoint of an operator between Hilbert spaces, selfadjoint operators, the Hilbert–Schmidt theorem.

Seminars

Exercise sheets

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.

Midterm program (05.11.2020)

Exam program (24.12.2020)

Contacts and availability of the instructors and TA's

Grading rules

References