Section: Course record | FUNCTIONS THEORY 2026-2027 - SCQ0094119

Select activity L1 (28 sept.) Course presentation. Spaces of continuous functions and Ascoli Arzel\`a compactness thm

L1 (28 sept.) Course presentation. Spaces of continuous functions and Ascoli Arzel\`a compactness thm File
Select activity L2 (30 sept): continuous and compact embeddings. The space of test functions. Test functions are dense in L^p, p

L2 (30 sept): continuous and compact embeddings. The space of test functions. Test functions are dense in L^p, p File
Select activity L3 (5 oct): DuBois Reymond lemma. Geodesic distance.

L3 (5 oct): DuBois Reymond lemma. Geodesic distance. File
Select activity L4 (7 oct): sets of class C^k. The outer normal. Partition of unit. Divergence thm

L4 (7 oct): sets of class C^k. The outer normal. Partition of unit. Divergence thm File
Select activity L5 (8 oct): integration on spheres formula. Computation of the volume of the ball and area of the sphere. Introduction to measure theory.

L5 (8 oct): integration on spheres formula. Computation of the volume of the ball and area of the sphere. Introduction to measure theory. File
Select activity L6 (12 oct): Absolutely continuous measures and singular measures. Lebesgue decomposition of measures. Measure theoretic boundary. Recap Riesz thm and weak compactness thms (Kakutani, Banach Alaoglu)

L6 (12 oct): Absolutely continuous measures and singular measures. Lebesgue decomposition of measures. Measure theoretic boundary. Recap Riesz thm and weak compactness thms (Kakutani, Banach Alaoglu) File
Select activity L7 (14 oct): Riesz theorems for Radon measure and weak star compactness. Application to L^1. (no proofs)

L7 (14 oct): Riesz theorems for Radon measure and weak star compactness. Application to L^1. (no proofs) File
Select activity L8 (15 oct): Hausdorff measures basic definition and properties. H^1=L^1 in R. Hausdorff dimension of a set. Example of the Cantor (ternary) set in R.

L8 (15 oct): Hausdorff measures basic definition and properties. H^1=L^1 in R. Hausdorff dimension of a set. Example of the Cantor (ternary) set in R. File
Select activity L9 (19 oct): Steiner symmetrization of a set. Isodiametric inequality in R^n. Proof that H^n=H^n_d=Lebesgue in R^n.

L9 (19 oct): Steiner symmetrization of a set. Isodiametric inequality in R^n. Proof that H^n=H^n_d=Lebesgue in R^n. File
Select activity L10 (21 oct): area formula and rectifiable sets. Definition of distributions and order of distributions.

L10 (21 oct): area formula and rectifiable sets. Definition of distributions and order of distributions. File
Select activity L11 (22 oct): support of a distribution. Principal value distribution. Convergence in the sense of distributions. Convolution between distributions and test functions. Density of C^\infty_c(U) in the space of distributions (no proof).

L11 (22 oct): support of a distribution. Principal value distribution. Convergence in the sense of distributions. Convolution between distributions and test functions. Density of C^\infty_c(U) in the space of distributions (no proof). File
Select activity L12 (2nov): derivative in the sense of distributions. Case of piecewise constant functions. Cantor Vitali function.

L12 (2nov): derivative in the sense of distributions. Case of piecewise constant functions. Cantor Vitali function. File
Select activity Cantor Vitali function (Devil's staircase)

Cantor Vitali function (Devil's staircase) File
Select activity L13 (4 nov): cantorian measures. Definition of weak derivatives. Fundamental solutions. Fundamental solution of the Laplacian.

L13 (4 nov): cantorian measures. Definition of weak derivatives. Fundamental solutions. Fundamental solution of the Laplacian. File
Select activity L14 (5 nov): fundamental solutions. Hypoelliptic differential operators. Weyl lemma: the harmonic distributions are smooth.

L14 (5 nov): fundamental solutions. Hypoelliptic differential operators. Weyl lemma: the harmonic distributions are smooth. File

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