Lectures log
First week (week 34)
- Tuesday: Naive set theory, functions. Pages 1–4 in Heil's book. Notes by a student physically present.
- Thursday: Cardinality, countable and uncountable sets. Pages 4–10 in Heil's book. Notes by a student physically present.
Second week (week 35)
- Tuesday: Suprema and infima. Pages 10–20 in Heil's book. Notes by a student physically present.
- Thursday: Vector spaces, span and independence. Pages 20–30 in Heil's book. Notes by a student physically present.
Third week (week 36)
- Tuesday: Metric spaces, convergence in metric spaces. Pages 31–43 in Heil's book. Notes by a student physically present.
- Thursday: Convergence and completeness in metric spaces. Pages 43–48 in Heil's book. Notes by a student physically present.
Fourth week (week 37)
- Tuesday: Completeness in metric spaces, open and closed sets, open balls, accumulation points and boundary points. Pages 48–64 in Heil's book. Notes by a student physically present.
- Thursday: Accumulation and boundary points. Closure, density and separability. Pages 64–70 in Heil's book. Notes by a student physically present.
Fifth week (week 38)
- Tuesday: Compact metric spaces. Continuity, uniform continuity and Lipschitz continuity. Fixed points. Pages 75–77 + 83–87 in Heil's book. Notes by a student physically present.
Sixth week (week 39)
- Tuesday: The Picard-Lindeöf Theorem and Picard iteration. Norms and normed spaces, l^p spaces, Hölder's inequality. Pages 99–111 in Heil's book. Notes by a student physically present.
- Thursday: The induced metric, Banach spaces, uniform convergence. Pages 111–122 in Heil's book. Notes by a student physically present.
Seventh week (week 40)
- Tuesday: Completeness of C[a,b]. Equivalent norms. Pages 124–139 in Heil's book. Notes by a student physically present.
- Thursday: Infinite series in normed spaces. Closed span and complete sequences, Schauder bases. Weierstrass' approximation theorem. Definition of inner products. Pages 140–147, 158–165, 191–192, and Theorem 4.6.2 in Heil's book. Notes by a student physically present.
Eighth week (week 41)
- Tuesday: Properties of the inner product. Hilbert spaces, examples. Pages 193–200 in Heil's book. Notes by a student physically present.
- Thursday: Orthogonality, orthogonal complements, orthogonal projections and the closest point theorem. Pages 201–207 in Heil's book. Notes by a student physically present.
Ninth week (week 42)
- Tuesday: Orthogonal and orthonormal sequences, orthonormal bases. Pages 208–216 in Heil's book. Notes by a student physically present.
- Thursday: Orthonormal bases continued, the Gram-Schmidt procedure, the complex trigonometric system. Pages 216–225 and 236–241 in Heil's book. Notes by a student physically present.
Tenth week (week 43)
- Tuesday: Linear operators on normed spaces, bounded operators, operator norm. Pages 243–255 in Heil's book. Notes by a student physically present.
- Thursday: Boundedness and continuity for linear operators, the space B(X,Y), isometries. Pages 255–265 in Heil's book. Notes by a student physically present.
Eleventh week (week 44)
- Tuesday: Isometries and isomorphisms, dual spaces and Riesz Representation Theorem, definition and existence of adjoint operators. Pages 264–271 and 279–283 in Heil's book, and the beginning of this note.
Twelfth week (week 45)
- Tuesday: The space B(X,Y) for finite-dimensional spaces X and Y, eigenvalues and eigenvectors, diagonalization of matrices. See pages 2–8 in this note.
- Thursday: Similarity transformations and Schur's lemma. The spectral theorem. Positive definite matrices. See pages 6–13 in this note.
Thirteenth week (week 46)
- Tuesday: Singular value decomposition, Polar decomposition. Notes from the zoom meeting.
- Thursday: Polar decomposition, the pseudoinverse. See the final pages of this note
Fourteenth week (week 47)
- Tuesday: Overview of the curriculum up to (but not including) Ch 6 in the book.
- Thursday: Completed overview of the curriculum. Covered problem 6 from Kont 2015, as well as problems 2 and 4 from Kont 2018. Notes from the curriculum review and the exam problems.