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 monotne 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 The space \(L^p(\Omega) \) for \( 1\le p \lt + \infty \) is separable (theorem 4.13). |
| Week 10 (17.11 and 21.11) | ... |
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 | ... |
| Exercise sheet 9 | Friday 21.11 | Submission link | ... |
| 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 |