Folland — Prerequisite Checklist

Folland — Prerequisite Checklist

  • TODO: The dependencies below have not been confirmed as completed in the existing notes and are not assigned new week numbers.
  • TODO: Before the relevant presentation, check your textbook and personal notes, then fill in source locations and proof dependencies below.

Before Week 05 — Duality of Lp

  • TODO: Folland §§3.1–3.3 — signed measures; Hahn / Jordan decomposition; Lebesgue–Radon–Nikodym Theorem; complex measures; total variation
  • TODO: Folland §§5.1–5.2 — bounded linear functionals; dual norm; Hahn–Banach background
  • TODO: Folland §6.1 — simple-function density; completeness; Hölder inequality
  • TODO: finite-measure localization; sigma-finite exhaustion; arbitrary-measure globalization
  • TODO: existing source links / theorem hypotheses / proof readiness

Before Week 06 — Endpoints and Consequences

  • TODO: Folland §§5.2, 5.4 — bidual; reflexivity; weak and weak-star topology
  • TODO: semifinite / sigma-finite / decomposable measure distinctions

Before Weeks 07–11 — Inequalities and Interpolation

  • TODO: Folland §2.5 — product measures; Fubini–Tonelli
  • TODO: maximum modulus principle; Hadamard Three-Lines Lemma

Before Weeks 12–16 — Radon Measures

  • TODO: Folland §§1.4, 4.4–4.5 — Carathéodory construction; compact Hausdorff / LCH spaces; Urysohn lemma; partitions of unity
  • TODO: Folland §3.3 — complex measures; total variation
  • TODO: Folland §5.4 — weak-star topology

Before Weeks 17–27 — Fourier Analysis

  • TODO: multi-indices; Taylor formula; smooth cutoff functions
  • TODO: Folland §3.5 — bounded variation
  • TODO: Folland §5.3 — Uniform Boundedness Principle
  • TODO: Folland §5.5 — Hilbert-space completeness; orthonormal expansions

Before Weeks 28–35 — Distributions and Sobolev Spaces

  • TODO: Folland §5.4 — locally convex topology; seminorms; continuous dual
  • TODO: distributional pairing versus Hilbert inner product conventions
  • TODO: Distinguish the Hs(Rn) setting of Folland §9.3 from the study of general W1p(Ω) spaces.

Before Weeks 36–42 — Probability

  • TODO: Chapters 1–3; §7.4 product measures; §8.6 measure transforms
  • TODO: Do not mark every topic in Chapters 8–9 as a required prerequisite for probability.

Before Weeks 43–48 — Further Measures and Integration

  • TODO: Chapter 4 topology; §7.1 Radon representation; §7.4 products
  • TODO: §1.4 outer measures; metric coverings; contraction mapping theorem
  • TODO: manifold charts; multivariable change of variables; Jacobians; Gram determinants
  • TODO: Before presenting §11.4, review differential geometry: densities, Riemannian metrics, top-degree forms, orientation, and Lie groups.

Cross-reference

  • Linear Maps
  • Inner Product Geometry
  • Matrix and Operator Norms
  • Folland Week 05