Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH. In particular the content of the courses Topology, Analysis 3 (measure and integration) as well as Analysis 4 (Introduction to Hilbert Spaces) will be assumed during the class.
| Week | Summary |
|---|---|
| Week 1 (18.09) |
Theorem 1.1 (Helly, Hahn-Banach analytic form). Definition of partially and totally ordered sets,
upper-bounds, maximal elements and inductive partially ordered sets. Axiom of choice. Zorn lemma 1.1 (without proof). Proof of Hahn-Banach theorem 1.1. Definition of the dual space to a real normed vector space. Dual spaces to real normed vector spaces are complete (i.e. define Banach spaces). Corollary 1.2 about the existence of the extension of any continuous linear form defined on a sub-vector space by a continuous linear form to the whole space without increasing the norm. |
| Week 2 (22.09 and 25.09) |
Corollary 1.3 about the existence for any vector in a normed space of an element in the dual whose norm is the same as the one of the vector and whose action on the vector is the square of the norm of this given vector. Uniqueness of the form does not hold in corollary 1.3. Uniqueness hold if the space is Hilbert (Corollary of Riesz-Fréchet theorem). Corollary 1.4 about the characterization of the length of a vector through the action of the elements in the dual. Definition of an hyperplane. In a normed vector space an hyperplane is closed if and only if the Linear form defining this hyperplane is continuous (Proposition 1.5). Definition of the separation of two disjoined sets by an Hyperplane (broad and strict versions). Hahn Banach theorem - first geometric form - about the separability of two disjoined convex sets, one of the two being open, by a closed hyperplane (theorem 1.6). The construction of a gauge characterizing a given open convex set (lemma 1.2). Hahn Banach first geometric form for a pair given by an open convex set and a point (lemma 1.3). The proof of Hahn Banach first geometric form. Hahn Banach second geometric form about the strict separability by a closed hyperplane of two disjoined closed convex sets, one of them being compact (theorem 1.7). The proof of Hahn Banach second geometric form. A corollary of Hahn Banach second geometric form about the characterization of the non density of a sub-vector space via the existence of a non zero element of the dual vanishing on this subvector space (corollary 1.8). The statement of Baire theorem (theorem 2.1). |
| Week 3 (29.09 and 02.10) | Proof of Baire's theorem. Banach Steinhaus Theorem on the uniform boundedness principle
(theorem 2.2). Corollary on the convergence and the uniform boundedness of sequences of continuous linear maps between two Banach spaces and whose action on every vector is converging (corollary 2.3). The characterization of bounded set in a Banach space through the boundedness of the action of any element of the dual (corollary 2.4). The ''dual formulation'' of corollary 2.4 (corollary 2.5). The open mapping theorem (theorem 2.6). The inverse of a bijective continuous linear map between two Banach spaces is continuous (corollary 2.7). If a vector space is Banach for two different norms and if the first norm is dominating the second one then the second one is dominating the first one ( corollary 2.8). The closed graph theorem (theorem 2.9). |
| Week 4 (06.10 and 10.10) | Various reminders from the course in topology. The definition of coarser and finer topologies. The existence of the coarsest topology making a family of maps into topological sets continuous (lemma 3.1). The characterisation of the convergence of sequences in this topology using the generating maps (Proposition 3.1). The characterisation of the continuity of a map from an arbitrary topological set into this coarsest topology using the generating maps (Proposition 3.2). The definition of the weak topology \( \sigma(E,E^\ast) \) of any given Banach space. Consequence of Hahn Banach second geometric form : the weak topology \( \sigma(E,E^\ast) \) is Hausdorff (Proposition 3.3). The Filter Basis of neighborhoods of \( \sigma(E,E^\ast) \) generated by finite families of elements in \( E^\ast \) (proposition 3.4). Properties on weakly converging sequences (proposition 3.5). The weak and the strong topologies coincide if \( E \) is finite dimensional (proposition 3.6). The weak closure and the sequential weak closure of some sets in infinite dimensional Banach spaces do not always coincide. The weak topology \( \sigma(E,E^\ast) \) of an infinite dimensional Banach space is never first countable. |
| Week 5 (13.10 and 17.10) | The closure of the unit sphere of an infinite dimensional Banach space for the weak topology \( \sigma(E,E^\ast) \) is equal to the closed unit ball of this space ( section 3.2 example 1). The open unit ball of an infinite dimensional Banach space is never open for the weak topology \( \sigma(E,E^\ast) \). Hence in infinite dimension the weak topology \( \sigma(E,E^\ast) \) has fewer open sets (section 3.2 example 2) than the strong topology issued from the norm. The dual of \( l^1 \) is \( l^\infty \). A sequence converges weakly in \( l^1 \) if and only if it converges strongly (section 3.2 remark 4). A convex set of a Banach space is closed for the weak topology \( \sigma(E,E^\ast) \) if and only if it is closed for the strong topology (Theorem 3.7). A convex function on a Banach space which is lower semi continuous for the strong topology is lower semi continuous for the weak topology (Corollary 3.9). A linear map between two Banach spaces \(E\) and \(F\) is continuous for the strong topology if and only if it is continuous for the weak topologies \(\sigma(E,E^\ast)\) and \(\sigma(F,F^\ast)\). Definition of the canonical injection \( J \) from \(E\) into \((E^\ast)^\ast\). The map \(J\) is an isometry. Definition of the ''weak star'' topology \(\sigma(E^\ast,E)\). The weak star topology \(\sigma(E^\ast,E)\) is Hausdorff (Proposition 3.11). The Basis of neighborhoods of \(\sigma(E^\ast,E)\) generated by finite subsets of \(E\) (Proposition 3.12). Various properties attached to the weak star convergence of sequences of \(E^\ast \) (Proposition 3.13). |
| Week 6 (20.10 and 24.10) | The closed unit ball of the dual space to a Banach space is compact for the weak star topology \( \sigma(E^\ast, E) \) (Theorem 3.16 Banach-Alaoglu-Bourbaki). Reminders from the topology course : 1) the definition of the cartesian product topology of an arbitrary family of topological spaces, 2) three characterisations of compact sets among which a set \(K \) is compact if any family of relative closed subsets of \( K \) satisfying the finite intersection property has a non empty intersection (in \( K \)). The cartesian product of an arbitrary family of compact topological spaces is compact (Tychonoff theorem). Proof of Tychonoff theorem (assuming the axiom of choice : an arbitrary cartesian product of non empty sets is non empty). Proof of Banach-Alaoglu-Bourbaki theorem. Any linear map from the dual \( E^\ast \) of an arbitrary Banach space \( E \) which is continuous for the weak star topology \( \sigma(E^\ast,E) \) is represented by an element in the image of the canonical injection \(J\) of \(E\) into \((E^\ast)^\ast \) (proposition 3.14). Characterisation of the closed Hyperplanes of \( E^\ast \) which are closed for the weak star topology \(\sigma(E^\ast,E)\) through the image of \(E\) by the canonical injection \(J\) of \(E\) into \((E^\ast)^\ast\) (corollary 3.15) |
| Week 7 (27.10 and 31.10) | Definition of Reflexive Banach spaces. A Bananch space is reflexive if and only if its unit closed ball for the norm is compact for the weak topology \( \sigma(E,E^\ast) \) (Kakutani thorem 3.17). The image of the unit closed ball of a Banach space by the canonical isometric injection \( J \) of \(E \) into the bidual \(E^{\ast\ast} \) is dense in the unit closed ball of \( E^{\ast\ast} \) with respect to the weak star topology \( \sigma(E^{\ast\ast},E^\ast) \) (Goldstine lemma 3.4). Proof of Kakutani theorem using Banach Alaoglu Bourbaki theorem and Goldstine lemma. Any closed sub-vector space of a reflexive Banach space is reflexive (proposition 3.20). A Banach space \(E \) is reflexive if and only if its dual space \(E^\ast \) is reflexive (corollary 3.21). Every bounded convex closed subspace of a reflexive Banach space is compact for the weak topology \( \sigma(E,E^\ast) \)(corollary 3.22). A convex lower semicontinuous function on a closed convex subset \( A \) of a reflexive Banach space \(E \) which is converging to infinity at infinity achieves its minimum in \( A \) (corollary 3.23). Definition of separable metric spaces. A subset of a separable metric space is separable (Proposition 3.25). A Banach space whose dual is separable is separable (Theorem 3.26). A Banach space is reflexive and separable if and only if its dual is reflexive and separable (Corollary 3.27). The restriction to the unit ball of the weak star topology of the dual \( E^\ast \) of a separable space \( E \) is metrizable (first part of theorem 3.28). |
| Week 8 (03.11 and 07.11) | If the closed unit ball of the dual of the Banach space \(E \) is metrizable then \( E \) is separable (second part of theorem 3.28). The unit ball of a Banach space is metrizable for the topology \( \sigma(E,E^\ast) \) if and only if its dual space \( E^\ast \) is separable (theorem 3.29). A uniformly bounded sequence of a dual space of a separable space has a subsequence which is converging for the weak star topology (Corollary 3.30). Any uniformly bounded sequence of a reflexive space has a subsequence which is converging for the weak topology (theorem 3.18). Definition of a uniformly convex space. A uniformly convex space is reflexive (Milman Pettis theorem 3.31). A weakly converging sequence in a uniformly convex Banach space is strongly converging if and only if the norm of the sequence is converging to the norm of the limit. Reminders from Analysis 3 : Beppo Levi monotone convergence theorem 4.1, dominated convergence theorem 4.2, Fatou lemma 4.1, continuous compactly supported functions are dense in \( L^1({\mathbb R}^n) \), Fubini theorem 4.5. Definition of \( L^p(\Omega) \) where \( \Omega \) is an open set of \( {\mathbb R}^n \). Hölder inequality (theorem 4.6). The space \( L^p \) defines a normed vector space (theorem 4.7). \( L^p \) is a Banach space (Fischer-Riesz theorem 4.8). |
| Week 9 (10.11 and 14.11) | A strongly converging sequence in \( L^p(\Omega) \) has a subsequence which converges almost everywhere and uniformly bounded by a fixed function in \( L^p \) (theorem 4.9). The spaces \( L^p(\Omega) \) for \( 1 \lt p \lt + \infty \) are reflexive (theorem 4.10). The first Clarkson inequality and the uniform convexity of \( L^p(\Omega) \) for \( 2\le p \lt +\infty \). The Riesz representation theorem for \(L^p \) spaces for \( 1 \lt p \lt +\infty \) (theorem 4.11). The second Clarkson inequality (with proof) and the uniform convexity of \(L^p(\Omega) \) for \(1\lt p \le 2 \). Consequence of the uniform convexity of \(L^p \) for \( 1 \lt p \lt +\infty \) : every bounded sequence in \( L^p \) admits a subsequence which converges weakly and it converges strongly if and only if the norm of the sequence converges to the norm of the limit. The space \(L^p(\Omega) \) for \( 1\le p \lt + \infty \) is separable (theorem 4.13). |
| Week 10 (17.11 and 21.11) | Proof of the separability of \( L^p \) (theorem 4.13). Riesz representation theorem for \( (L^1(\Omega))^\ast \) : \( (L^1(\Omega))^\ast= L^\infty(\Omega) \) (theorem 4.14). \( L^1(\Omega) \) is not reflexive. Example of a continuous linear form on \( (L^\infty(\Omega))^\ast \) which cannot be represented by an \( L^1 \) function. The closed unit ball in \(L^\infty(\Omega) \) is compact for the weak star topology and the restriction of the topology \( \sigma(L^\infty,L^1) \) to this unit ball is metrisable. Consequence : every uniformly bounded sequence in \(L^\infty(\Omega) \) admits a subsequence which weak star converges. A Banach space which posses an uncountable union of disjoint non empty family of open sets is non separable (Lemma 4.2). The space \( L^\infty(\Omega) \) is not separable. First Young inequality (theorem 4.15). The support of a convolution of an \(L^1 \)and \(L^p \) function is included in the closure of the sum of the supports of the two functions (proposition 4.18). The convolution between a compactly supported continuous function and an \(L^1_{loc} \) function is continuous (proposition 4.19). The convolution between a compactly supported \( C^k \) function and an \( L^1_{loc} \) function is \(C^k \) (proposition 4.20). |
| Week 11 (24.11 and 28.11) |
Proof of proposition 4.20. Definition and explicit construction of mollifiers. The strong approximability of \( L^p \) functions by their convolutions with mollifiers (proposition 4.21 and theorem 4.22). The density of smooth compactly supported functions in \(L^p \) (corollary 4.23). Definition of Sobolev Spaces \(W^{1,p}(\Omega)\) where \(\Omega\) is an open subset of \({\mathbb R}^n \)and \(1\le p\le +\infty\) as well as the Sobolev norm \(W^{1,p}\). The Sobolev spaces defines Banach spaces. They are reflexive if \( 1 \lt p \lt +\infty \) and separable if \( 1\le p \lt +\infty \) (proposition 9.1). The space of smooth compactly supported functions is dense in \( W^{1,p}({\mathbb R}^n)\) (Friedrich theorem 9.2 for \(\Omega={\mathbb R}^n \) ). The characterisation of Sobolev spaces \(W^{1,p}({\mathbb R}^n) \)using finite differences (Proposition 9.3 for \(\Omega={\mathbb R}^n\)). |
| Week 12 (01.12 and 05.12) |
Proof of the characterization of Sobolev functions in \(W^{1,p}({\mathbb R}^n) \) using finite differences (proposition 9.3).
The Leibnitz Rule in \(L^\infty\cap W^{1,p}({\mathbb R}^n) \) (proposition 9.4). The Chain rule for the composition of a \( C^1 \) function in \( W^{1,\infty}({\mathbb R}) \) with a Sobolev fuction in \( W^{1,p}({\mathbb R}^n) \) (Proposition 9.5). The Sobolev embedding theorem of \( W^{1,p}({\mathbb R}^n) \) into \( L^{p^\ast}({\mathbb R}^n) \) when \( 1\le p \lt n \) (Theorem 9.9 of Sobolev Gagliardo Nirenberg). Gagliardo lemma 9.4. The continuous canonical injection from \( W^{1,p}({\mathbb R}^n)\) into \( L^q({\mathbb R}^n) \) for \(p \lt n \) and \( p\le q\le p^\ast \) (Corollary 9.10). The continuous canonical embedding of \( W^{1,n}({\mathbb R}^n) \) into \( L^q({\mathbb R}^n) \) for \( p\le q \lt +\infty \). A counter exemple of the embedding of \( W^{1,2}({\mathbb R}^2) \) into \( L^\infty({\mathbb R}^2) \). The Morrey Sobolev embedding of \( W^{1,p}({\mathbb R}^n) \) into \( C^{0,\alpha}({\mathbb R}^n) \) for \( p \gt n \) and \( \alpha=1-n/p \) (Theorem 9.12). |
| Week 13 (08.12 and 12.12) | The \(L^\infty\) bound in the Morrey-Sobolev embedding. Arzela Ascoli theorem 4.25. Proof of Arzela-Ascoli using Tychonoff theorem. The restriction of the coarsest cartesian product topology (Tychonoff topology) on an equicontinuous bounded family of the space of continuous functions of a compact metric space coincides with the strong metric topology. Theorem of Riesz-Fr\'echet-Kolmogorov on the compactness in \(L^p\) of bounded subsets of \(L^p(\Omega)\) \( (\Omega\) bounded open subset of \( {\mathbb R}^n\)) satisfying uniform \(L^p \) convergence to zero of the differences of each functions with its infinitesimal translations (theorem 4.26 and corollary 4.27). Rellich Kondrachoff theorem on the compactness of the canonical inclusion map from \(W^{1,p}({\mathbb R}^n) \) into \( L^q(\Omega) \) where \( \Omega \) is a bounded open set, (\(p \lt n\) and \(1\le q \lt p^\ast \) or \( p=n \) and \( 1\le q \lt +\infty \)) and the case \( p>n\) (theorem 9.16). Introduction to Fredholm Theory. The definition of the dual operator of a bounded linear operator between two normed vector spaces. The dual operator is bounded and its norm equals the norm of the operator (application of Hahn Banach). Some fundamental properties of dual and bi-dual operators. The definition of the Hilbert adjoint of a bounded operator between an Hilbert space and itself. The definition of self-adjoint/symmetric operators in Hilbert spaces. An example of a self-adjoint operator in \(l^2 \). |
| Week 14 (15.12 and 19.12) |
A subvector space is dense in a Banach space if and only its orthogonal is zero. A subvector space of the dual of a Banach space is weak star dense in the dual if and only if the pre-orthogonal is zero. A continuous linear operator between Banach spaces has a dense image if and only its dual is injective. The operator is injective if and only if the dual has a weak star dense image. The Douglas factorization theorem. The normed vector space defined by the quotient of a Banach space with a closed sub-vector space is Banach. The adjoint of the projection map from this Banach space to this quotient is an isometry. The image of the dual of the projection map equals the orthogonal to the closed sub-vector space. The closed range theorem: a bounded linear operator between two Banach spaces has a closed image if and only if its dual operator has a closed image. |
The new exercises will be posted here on Fridays. We expect you to look at the problems over the weekend and to prepare questions for the exercise class on Monday.
Please hand in your solutions by the following Friday until 12:00 at lastest online via the following submission link.
Make sure that your solution is one PDF file and that its file name is formatted in the following way:
Office hours: Office hours will be held by Gemei Liu every Tuesdays from 14:00 to 15:00.
During these hours, she will be available in her office (HG F 28.3) for any questions about the lecture material or exercise sheets.
| Exercise sheet | Due by | Upload link | Solutions |
|---|---|---|---|
| Exercise sheet 1 | Friday 26.09 | Submission link | Solution sheet 1 |
| Exercise sheet 2 | Monday 06.10 | Submission link | Solution sheet 2 |
| Exercise sheet 3 | Friday 10.10 | Submission link | Solution sheet 3 |
| Exercise sheet 4 | Friday 17.10 | Submission link | Solution sheet 4 |
| Exercise sheet 5 | Friday 24.10 | Submission link | Solution sheet 5 |
| Exercise sheet 6 | Friday 31.10 | Submission link | Solution sheet 6 |
| Exercise sheet 7 | Friday 07.11 | Submission link | Solution sheet 7 |
| Exercise sheet 8 | Friday 14.11 | Submission link | Solution sheet 8 |
| Exercise sheet 9 | Friday 21.11 | Submission link | Solution sheet 9 |
| Exercise sheet 10 | Friday 28.11 | Submission link | Solution sheet 10 |
| Exercise sheet 11 | Friday 05.12 | Submission link | Solution sheet 11 |
| Exercise sheet 12 | Friday 12.12 | Submission link | Solution sheet 12 |
| Exercise sheet 13 | Friday 19.12 | Submission link | Solution sheet 13 |
| time | room | assistant |
|---|---|---|
| Mo 09:15-10:00 | HG G 26.1 | Stefano Decio |
| Mo 09:15-10:00 | HG G 26.5 | Antoine Detaille |