Lectures log
First week (week 34)
- Tuesday: Naive set theory, functions, cardinality. Pages 1–6 in Heil's book.
- Wednesday: Cardinality, infima and suprema. Pages 6–14 in Heil's book.
Second week (week 35)
- Tuesday: Vector spaces, span and independence. Sections 1.10-1.11 in Heil's book.
- Wednesday: Metrics, convergence and Cauchy sequences. Sections 2.1-2.2 in Heil's book.
Third week (week 36)
- Tuesday: Convergence and completeness in metric spaces. Section 2.2 in Heil's book.
- Wednesday: Open and closed sets, accumulation points and boundary points, closure and density. Sections 2.3-2.6 in Heil's book.
Fourth week (week 37)
- Tuesday: Separability. Compact sets in metric spaces. Continuity of functions in metric spaces. Sections 2.6, 2.8 and 2.9 in Heil's book.
- Wednesday: Banach's fixed point theorem and application. See this note
Fifth week (week 38)
- Tuesday: Picard-Lindelöf theorem and Picard iteration. Norms and normed spaces, l^p spaces. See sections 3.1-3.2 in Heil's book.
- Wednesday: The induced norm, Banach spaces, uniform convergence of functions. See sections 3.3-3.5 in Heil's book.
Sixth week (week 39)
- Wednesday: Equivalent norms, selected topics from chapter 4. See sections 3.6–3.7 and 4.1–4.2
- Thursday: Span, closed span and complete sequences. Schauder bases, Weierstrass approximation theorem. Properties of the inner product. See sections 4.4–4.6 and 5.1–5.2 in Heil's book.
Seventh week (week 40)
- Wednesday: Inner product spaces, Hilbert spaces, examples. Orthogonal and orthonormal sets and sequences, orthogonal complements. See sections 5.3–5.5 in Heil's book.
- Thursday: Orthogonal projections and the Closest Point Theorem. Orthonormal sequences. See sections 5.6-5.7 in Heil's book.
Eighth week (week 41)
- Plan: Orthonormal bases, Gram-Schmidt's orthogonalization procedure, the complex trigonometric system. Linear operators on normed spaces, bounded operators. See sections 5.8–5.9, 5.11, 6.1–6.2 in Heil's book.
- Wednesday: Orthonormal bases, Gram-Schmidt's orthogonalization procedure. See sections 5.8–5.9 in Heil's book.
- Thursday: The complex trigonometric system, linear operators on normed spaces. See sections 5.11 and 6.1 in Heil's book.
Ninth week (week 42)
- Plan: bounded operators, equivalence of bouded and continuous linear operators, the space B(X,Y). Isometries and isomorphisms. Dual space and Riesz representation theorem. See sections 6.2–6.6 and 6.8 in Heil's book.
- Wednesday: Bounded operators, equivalence of bounded and continuous linear operators. See sections 62–6.4 in Heil's book.
- Thursday: The space B(X,Y). Isometries and isomorphisms. See sections 6.5–6.6 in Heil's book.
Tenth week (week 43)
- Wednesday: Dual space and Riesz Representation Theorem. Existence of adjoint operators and examples. See section 6.8 in Heil's book and the note on adjoints.
- Thursday: Properties of adjoint operators. Normal, unitary and self-adjoint operators. Bounded linear operators between finite-dimensional spaces. See the note on adjoints and that on topics in linear algebra.
Eleventh week (week 44)
- Wednesday: Rank-nullity theorem. Diagonalizations and similar matrices. See pages 3–8 in this note.
- Thursday: Similar matrices and applications to linear systems of differential equations. Schur's triangulation lemma. See pages 8–10 in this note.
Twelfth week (week 45)
- Wednesday: Spectral theorem for normal matrices. Positive-definite matrices and singular value decomposition. See pages 10–14 in this note
- Thursday: Singular value decomposition, examples and consequences. Pseudoinverse. See pages 14–19 in this note
Thirteenth week (week 46)
- Wednesday: The pseudoinverse, and its application to consistent and inconsistent linear systems of equations. Start of curriculum review.
- Thursday: Curriculum review.
Fourteenth week (week 47)
- Wednesday: Curriculum review completed. Problems 1, 2 and 3a,b of the November exam of 2018
- Thursday: Problems 3c, 4, 5 and 6 of the November exam of 2018. Problems 1 and 2 from the corresponding make-up exam.