![]() ![]() ![]() Michel Willem, Functional Analysis, Fundamentals and Applications The main steps to prove Thms 4.1.1, 4.2.1, 4.2.2 and 4.2.3 should be known.Ī good reference for a constructivist approach to Hahn-Banach (i.e. We then proved two theorems of Krein and Milman (Thms 4.2.2 and 4.2.3), about extreme points of compact convex sets. The complex case is treated in the problems, as well as the separation theorem (Thm 4.2.1), of which we only proved part (b). We proved the Hahn-Banach extension theorem (Thm 4.1.1) in the case of a real vector space. September 22, François Genoud, room WN-M639. Haase, Functional Analysis: An Elementary Introduction, Chap. 3.5.1 and of Theorem 3.6.1(d).įurther recommended reading on Banach-Steinhaus are: We gave conditions for a TVS to be metrizable/normable. Students should know the proof in the Banach space setting (Thm 2.2.1) and are advised to compare it with the general one. We proved the Banach-Steinhaus theorem for general TVS in full detail. We covered Sections 3.2-3.6 of the lecture notes. September 15, François Genoud, room WN-M639. Warning: in Thm 2.1.1 it is essential that S be non-empty. You can read the proof of this theorem in the notes, I will skip it. ![]() We concluded by introducing the first basic properties of TVS, up to Thm 3.1.3 about linear maps. A more general proof for topological vector spaces (TVS) will follow. The latter we proved in the Banach space context. ![]() The main results we proved are Baire's theorem (Thm 2.1.1) and the Banach-Steinhaus theorem (Thm 2.2.1). September 8, François Genoud, room WN-M639. As always in mathematics, it is highly recommended that you work out as many problems as possible in order to become familiar with the various topics covered. Substantial exercise sets will be given each week (at least in the first few weeks, see the list of lectures below) so you can try yourself with the new material. Homework Assignment 1 consists of the following problems from the weekly exercise sets posted below: The homework assignments will become available below.Ī selection of particularly important problems, taken from the exercise sets (see below), will be given as Homework Assignment (HA) and graded. PLEASE NOTE THAT THERE IS NO RETAKE FOR THE HOMEWORK ASSIGNMENTS. Homework assignments (30%) and a written exam (70%). The final grade is based on the results obtained for the The lectures are scheduled on Tuesdays from 10:15 through 13:00 in the Mathematics and Sciences building (except October 6) of the Vrije Universiteit in Amsterdam, in room TBA.Ī map and the address of the building can be found here. The course is taught by François Genoud and Onno van Gaans. Information on the aim and topics of the course and the book by Rudin can be found on the mastermath page of the course. Make sure that you bring a photo ID with you to the exam! The retake is scheduled on TUESDAY February 2, 2016, 10:00-13:00, at the VU in Amsterdam, Mathematics and Sciences building ("W & N building"), room P663. There are construction works in this building, so you may have to follow a detour through the basement. For easy orientation, have a look at a picture of the VU campus and a small map of the building. Room MF-FG1 is lecture room FG1 (one of the main lecture rooms) of the building of the Medische Faculteit (next to the Mathematics and Sciences building), Van der Boechorststraat 7-9, Amsterdam. The written exam is on January 12, 2016, 10:00-13:00 at the VU in room MF-FG1. Or Marcel de Jeu if you are interested in participating. Please visit the website for more information (in particular on the credit points and grade) and contact Onno van Gaans In the Spring semester of 2016, there will be a functional analysis seminar at Leiden University. If you examine the standard proof of these factors for operator on Hilbert spaces, then they are rather similar.Mastermath Functional Analysis 2015 home page From this, we see that $T^*$ is densely-defined if and only if $T$ is closable. One can also reverse this, starting with a weak $^*$-closed operator $E_2^*\rightarrow E_1^*$. So if $E_1,E_2$ are reflexive, then $T^*$ is closed in the weak, and so norm, topology. $T^*$ is always closed in the weak $^*$-topology. $T^*$ is the graph of an operator when $(0,y^*)\in G(T^*)\implies y^*=0$, equivalently, when $T$ is densely defined. Identify $(E_1\oplus E_2)^*$ with $E_1^*\oplus E_2^*$ so the annihilator of $G(T)$ is In terms of the graph of the operators, this means that $(x^*,y^*)\in G(T^*)$ exactly when If $T: E_1 \supseteq D(T)\rightarrow E_2$ is a linear map between Banach spaces, then we define $x^*\in D(T^*)$ with $T^*(x^*)=y^*$ to mean that $y^*(x) = x^*(T(x))$ for each $x\in D(T)$. You can use essentially the same definition. ![]()
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