We. 21.08 | Introduction. Lecture notes sections 1.1–1.4 (Sets; Membership and inclusions; Set operations; Relations). | Friedberg et al. 549–550; Kreyszig 609–617. |

Fr. 23.08 | Lecture notes sections 1.4–1.6. (Relations; Invertibility; Vector spaces) | Friedberg et al: 6–12, Kreyszig 613–617, 50–53; Strang 69–71. |

We. 28.08 | Lecture notes sections 1.4–1.8. (Vector spaces; Normed spaces; Metric spaces) | Friedberg et al: 1.2; Kreyszig 2.1–2.2; Young 13–14; Strang 2.1. |

Fr. 30.08 | Lecture notes sections 1.8–1.10. (Metric spaces; Balls and spheres; Interior points, boundary points, open and closed sets.) | Kreyszig 1.1–1.3; Young Chapter 2 (the parts you understand). |

We. 04.09 | Lecture notes sections 1.10–1.11. (Interior points, boundary points, open and closed sets; Limits) | Kreyszig 1.4. |

Fr. 06.09 | Lecture notes sections 1.11–1.12. (Limits; Completeness) | Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. |

We. 11.09 | Lecture notes section 1.12 (Completeness) | Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. |

Fr. 13.09 | Lecture notes section 1.13 (Completions) | Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. |

We. 18.09 | Lecture notes sections 2.1–2.3. (Linear subspaces; linear dependence; Bases and dimension) | Friedberg et al: 1.3–1.6; Kreyszig 2.1 (2.4); Strang 2.1, 2.3. |

Fr. 20.09 | Lecture notes sections 2.3–2.5. (Bases and dimension; Schauder bases; Basis transformations) | Friedberg et al: 2.5; Kreyszig 2.3; Strang 2.1, 2.3. |

We. 25.09 | Lecture notes sections 2.5–2.6 (Basis transformations; Gaussian elimination) | Friedberg et al: 3.4 (especially 183–188); Strang 1.5, 2.2. |

Fr. 27.09 | Lecture notes sections 2.6–2.7 (Gaussian elimination; Linear transformations) | Friedberg et al: 2.1–2.2; Kreyszig 2.6; Strang 2.6. |

We. 02.10 | Lecture notes section 2.7 (Linear transformations): The vector space \(L(X,Y)\). Characterization of linear transformations on finite-dimensional vector spaces; matrix representations. Kernels and ranks. Characterization of injective linear transformations. | Friedberg et al: 2.1–2.4; Kreyszig 2.6 and 2.9 (not the part on functionals and duals); Strang 2.2, 2.4 and 2.6. |

Fr. 04.10 | Lecture notes section 2.7 (Linear transformations): null spaces, column spaces and row spaces; geometric interpretation of the nullspace. The rank–nullity theorem; reduction on \(\mathbb R^n\), and geometric interpretation. Summary on linear equations (the Fredholm alternative). |

We. 09.10 | Lecture notes section 2.8 (Bounded linear transformations): Boundedness. The operator norm and the normed space \(B(X,Y)\). Functionals, duals, and the Riesz representation theorem. \(B(X,Y)\) Banach for \(Y\) Banach. | Friedberg et al: 2.6; Kreyszig 2.10; Young: Chapters 6 and 7 (up to 7.2); Strang: 7.2 (the parts about the matrix norm). |

Fr. 11.10 | Lecture notes section 2.8 (Bounded linear transformations): Equivalence of boundedness and continuity for linear operators. Finite-dimensional linear mappings are bounded. Kernels of continuous mappings are closed. |

We. 16.10 | Solving differential equations: initial-value problems; formulation as first-order systems; Peano's theorem; Lipschitz continuity. | Kreyszig: Chapter 5. |

Fr. 18.10 | Contractions and the Banach fixed-point theorem; the Picard–Lindelöf theorem and Picard iteration. |

We. 23.10 | Constant-coefficient linear equations \(\dot x = Ax\). Eigenvalues, eigenvectors and the spectrum of a bounded operator. Characteristic polynomials: the spectrum of a matrix consists of its eigenvalues. Algebraic and geometric mulitplicity of eigenvalues. Characterization of the solution space of \(\dot x = Ax\). The exponential map for matrices and explicit solution formula. | With discernment: Friedberg: 5.1–5.2, 5.4, 6.6, 7.1–7.2.; Kreyszig: 7.1–7.2, Strang: Chapter 5 and Appendix B. |

Fr. 25.10 | Spectral decompositions: generalized eigenspaces. The Riesz index and a different characterization of algebraic multiplicity. Cayley–Hamilton and its consequences. The spectral decomposition, the Jordan form, and the spectral theorem for matrices. |

We. 30.10 | Inner-product spaces and their properties. The Cauchy–Schwarz inequality, parallelogram law and polarization identity. | Young: Chapters 1 and 3. |

Fr. 01.11 | Hilbert spaces. Convex sets and the closest point property. |

We. 06.11 | Orthogonal vectors and sets. The projection theorem (corollary: strict subspace characterization) and the Riesz representation theorem. | Young: Chapter 4 and Section 6.1 (Riesz representation theorem). |

Fr. 08.11 | Orthonormal sequences, Fourier coefficients and Fourer series. Bessel's inequality. Fourier coefficients are best possible (corollary: closest point). Convergence as an \(l_2\)-property. The Fourier series theorem. |

We. 13.11, Fr. 15.11 | Adjoints and decompositions. Self-adjoint operators and symmetric/Hermitian matrices. Unitary operators and unitary/orthogonal matrices. The spectral theorem for Hermitian/symmetric matrices. Positive definiteness. QR- (Gram–Schmidt), Cholesky- and singular value-decompositions. | Young: Sections 7.3 and 8.2. Friedberg: Sections 6.2–6.8, Strang: Sections 3.4, 5.2, 5.5, 6.1–6.3. |

We. 20.11, Fr. 22.11 | Repetition. | |