MATH 6320 - Fall 2024
Theory of Functions of a Real Variable
    Course Info. View syllabus. Office hours: PGH 641A, Tu 10-11am, We 1-2pm.

    MATH 4331/6312 - Fall 2023
    Introduction to Real Analysis I
      Course Info. View syllabus. Office hours: PGH 604, Tu 9:30-10:30am, We 1-2pm.

      MATH 7321 - Spring 2023
      Functional Analysis II
        Course Info. View syllabus. Office hours: PGH 604, Tu 1-2pm, We 2-3pm.
          Week 1. Course information, brief review of last term, see notes (Bernhard Bodmann). Connectedness, groups and Banach algebras, see notes (Bernhard Bodmann).
            Week 2. Commutative C*-algebras and the Gelfand transform, see notes (Cristian Meraz). Towards a functional calculus for operators, see notes (Dipanwita Bose).
              Week 3. Functional calculus and its limitations see notes (Kumari Teena). The commutant and its properties, see notes (Tanvi Telang).
                Week 4. Von Neumann's double commutant theorem, see notes (Drake Walker). Schur's lemma and its consequences for representations of involutive semigroups on infinite-dimensional Hilbert spaces, see notes (Cristian Meraz).
                  Week 5. Towards functional calculus for normal operators with measurable functions, see notes (Yerbol Palzhanov). Maximal abelian C*-algebras, see notes (Kumari Teena).
                    Week 6. Weak operator topology vs. weak-* topology on L∞(μ), see notes (Lukasz Krzywon). Cyclic vectors and how to get a measure for functional calculus with measurable functions, see notes (Cristian Meraz).
                      Week 7. Functional calculus for Borel measurable functions of normal operators on Hilbert spaces, see notes (Dipanwita Bose). Towards the Gelfand-Naimark-Segal representation theorem. Introduction to reproducing kernel Hilbert spaces, see notes (Sean Campbell).
                        Week 8. Reproducing kernel Hilbert spaces, see notes (Bernhard Bodmann). Relationships between the Hilbert spaces associated with reproducing kernels and the kernel functions, see notes (Bernhard Bodmann).
                          Week 9. Positive functionals and states, see notes (Manpreet Singh). Positivity and (positive) square roots of elements in a C*-algebra, see notes (Lukasz Krzywon).
                            Week 10. Properties of positive linear functionals on C*-algabras and states, see notes (Drake Walker). The cone of positive linear functionals and duality for cones, see notes (Tanvi Telang).
                              Week 11. Examples of the GNS construction. Cyclic representations and the GNS construction, see notes (Kumari Teena).
                                Week 12. From the GNS construction to an isometric isomorphism of C*-algebras, see notes (Yerbol Palzhanov). Irreducible representations and pure states in the GNS construction, see notes (Cristian Meraz).
                                  Week 13. Convex geometry of states and consequences, see notes (Manpreet Singh).

                                  MATH 7320 - Fall 2022
                                  Functional Analysis
                                    Course Info. View syllabus. Office hours: PGH 604, Tu 1:30-3pm, We 10:30-11:30am.
                                      Week 1. Course information, brief summary of content. Review of fundamental concepts, see notes, notes (Bernhard Bodmann). Check if you remember some fundamental theorems listed in this handout.
                                        Week 2. Hilbert spaces and operators on Hilbert spaces. Duality, Riesz representation theorem, summability, see notes (An Vu) and notes (Drake Walker). Direct sums of Hilbert spaces, see notes (Lukasz Krzywon).
                                          Week 3. Operators on Hilbert spaces. The adjoint map and its properties. Unitary, Hermitian, skew-Hermitian, and normal operators, see notes (Caleb Barnett) and notes (Manpreet Singh). Relationships between types of operators, hermitian and unitary versus normal operators. Characterization of unitary operators, see notes (Jessie McKim) and notes (Nick Fularczyk).
                                            Week 4. Unitaries as surjective isometries. Characterization of orthogonal projections, see notes (Alina Rajbhandari) and notes (Kumari Teena). Relationship between isometries and orthogonal projections. Spectral theory, see notes (Tanvi Telang) and notes (Dipanwita Bose).
                                              Week 5. Representations of involutive semigroups. Invariant subspaces. Cyclic representations, see notes (Cristian Meraz) and notes (Anshi Gupta). Decomposition of a non-degenerate representation into cyclic components, see notes (Sean Campbell) and notes (Manpreet Singh).
                                                Week 6. Non-degenerate representations, finite dimensional representations, and the case of abelian involutive semigroups, see notes (Lukasz Krzywon) and notes (An Vu). Characters and classification of the finite-dimensional representations of abelian involutive semigroups, see notes (Caleb Barnett).
                                                  Week 7. The spectral theorem in finite-dimensional Hilbert spaces. Banach algebras and spectral theory. From involutive algebras to C*-algebras. see notes (Nick Fularczyk) and notes (Jessie McKim). Properties of elements in involutive algebras, how to test the norm identity for a C*-algebra. see notes (Phuong Tran).
                                                    Week 8. Characters on Banach-*-algebras. Examples. Characterization of C*-algebras. see notes (Christian Meraz) and notes (Alina Rajbhandari). From characters on Z to Fourier series. see notes (Anshi Gupta) and notes (Dipanwita Bose).
                                                      Week 9. Characters of L1(Rd). What to do when a Banach algebra does not have a unit, see notes (Drake Walker) and notes (Tanvi Telang). Extending a Banach algebra by adjoining a unit, see notes (Kumari Teena) and notes (Phuong Tran).
                                                        Week 10. Properties of C*-algebras without unit when extending them. Properties of the spectrum for elements in Banach algebras see notes (Nick Fularczyk) and notes (An Vu). More properties of the spectrum and proofs by complex analysis, see notes (Christian Meraz) and notes (Jessie McKim).
                                                          Week 11. The spectrum is non-empty and compact, see notes (Lukasz Krzywon) and notes (Caleb Barnett). An asymptotic formula for the spectral radius, see notes (Alina Rajbhandari) and notes (Drake Walker).
                                                            Week 12. Relation between norm and spectral radius in C*-algebras, see notes (Kumari Teena) and notes (Tanvi Telang). Homomorphisms on Banach algebras, revisited, see notes (Phuong Tran) and notes (Dipanwita Bose).

                                                            MATH 6361 - Spring 2022
                                                            Applicable Analysis II
                                                              Course Info. View syllabus. Office hours: PGH 604, Tu 1:30-3pm, We 1-2pm.
                                                                Week 1. Course information, brief summary of content. Inner product spaces. Cauchy-Schwarz inequality. Norm induced by the inner product. Hilbert spaces.
                                                                  Homework Set 1, due January 27.
                                                                    Week 2. Best approximation in closed linear subspaces as a linear map: orthogonal projections. Orthonormal systems, Gram Schmidt procedure, orthonormal bases (Hunter/Nachtergaele Ch. 6).
                                                                      Homework Set 2, due February 3.
                                                                        Week 3. Fourier series and their relationship with an orthonormal basis of complex exponentials. Pointwise convergence of Fourier series. For more material, consult Davidson/Donsig Ch. 14.5 and 14.7. A brief summary with some illustrations is given in the presentation on Fourier series and orthonormal bases in inner product spaces.
                                                                          Homework Set 3, due February 10.
                                                                            Week 4. Weak convergence for sequences in a Hilbert space. The relationship between weak convergence and convergence with respect to the norm. Orthonormal bases as examples of weakly convergent sequences. Application: Decay of Fourier coefficients. For related material, see Hunter/Nachtergaele Ch. 8.6.
                                                                              Homework Set 4, due February 17.
                                                                                Week 5. Operators and sesquilinear forms on Hilbert spaces. Extracting information about an operator from its quadratic form.
                                                                                  Homework Set 5, due March 3.
                                                                                    Week 6. The Lax-Milgram theorem: Coercivity as a condition for invertibility of an operator. Application to the (weak) solution of a boundary value problem. Hilbert-Schmidt operators. Bounded self-adjoint operators. Compact operators. Eigenvalues and eigenvectors.
                                                                                      Homework Set 6, due March 24.
                                                                                        Week 7. The orthonormal basis of eigenvectors for compact self-adjoint operators. For more details, see Hunter/Nachtergaele, Ch. 9.
                                                                                          Homework Set 7, due March 31.
                                                                                            Week 8. The spectral theorem for compact normal operators.
                                                                                              Homework Set 8, due April 7.
                                                                                                Week 9 and 10. Separation theorems, from Hahn-Banach to separating hyperplanes. (For details, see the handout on MS Teams, Section 4. A good textbook reference is Barvinok, A Course in Convexity, Ch. III, up to Thm III.(3.2).)
                                                                                                  Homework Set 9, due April 28.

                                                                                                  MATH 6360 - Fall 2021
                                                                                                  Applicable Analysis
                                                                                                    Course Info. View syllabus. Office hours: PGH 604, Tu 1-2pm, We 1-2pm.
                                                                                                      Week 1. Review of metric spaces. Open and closed sets. Completeness. Characterization of compact sets by sequential compactness or by total boundedness and completeness. Definition of contraction mappings. (Related material can be found in Davidson/Donsig Chs. 9 and 11, but we discuss these mappings in the setting of complete metric spaces, while the book mostly treats Banach spaces). Uniform continuity. Examples of contraction mappings.
                                                                                                        Homework Set 1, due September 2. Solutions posted on MS Teams, in folder Class Materials.
                                                                                                          Week 2. Examples of contraction mappings. Integral operators as contractions. Solutions to integral equations. Solutions to ordinary differential equations. Contractivity of an integral operator related to initial value problems. Global Picard theorem on existence and uniqueness of solutions to initial value problems.
                                                                                                            Homework Set 2, due September 9.
                                                                                                              Week 3. Local Lipschitz continuity condition and existence and uniqueness of a solution on a subinterval. Stability of solutions. Towards Lp-spaces. A generalized arithmetic-geometric mean inequality and Hölder's inequality. See Ch. 7.7 for material.
                                                                                                                Homework Set 3, deferred to noon, September 17.
                                                                                                                  Week 4. Minkowski's inequality. lp spaces as normed spaces. Metric completions. Extending a uniformly continuous functions to the completion of its domain.
                                                                                                                    Homework Set 4, due September 23.
                                                                                                                      Week 5. Banach spaces, Schauder bases. Linear maps on Banach spaces.
                                                                                                                        Homework Set 5, due October 14.
                                                                                                                          Week 6. Operator norm. Dual spaces and completeness. The dual space of lp.
                                                                                                                            Homework Set 6, due October 21.
                                                                                                                              Week 7. Uniform boundedness and its implications for the convergence of Fourier series and polynomial interpolation.
                                                                                                                                Homework Set 7, due October 28.
                                                                                                                                  Week 8. Open mappings and bounded inverses.
                                                                                                                                    Homework Set 8, due November 4.
                                                                                                                                      Week 9. Closed subspaces and nearest vectors. Uniform convexity of Lp for 1<p<∞
                                                                                                                                        Homework Set 9, due December 2.
                                                                                                                                          Week 10. Finite dimensional normed spaces. Completeness. Isomorphisms. Equivalence of any pair of norms. Controlling the condition number of an isomorphism.