Preface Changes in Recent Years
Changes in 2024/2025.
- Last year during, during the semester, I added Figure 1.7 and Figure 1.8 illustrating Lemma 1.28. As usual, I also fixed typos and added new hints and solutions for exercises.
- I have replaced the notation \(L^2(a,b)\) inherited from previous years with \(L^2([a,b])\text{,}\) which is more consistent with the notation for the other function spaces introduced in Chapter 3. I am told that this is consistent with the Year 4 Analysis units.
- Convention 1.32 now explicitly lays out the standard notational conventions for sequences.
- I have promoted Proposition 1.47 on the continuity of vector space and inner product space operations from an exercise to a proposition. It is proved using Theorem 1.46 on sequential continuity, and the necessary facts about sequences are requested earlier in Exercise 1.3.1.
- An additional part has been added to Exercise B.2 on Problem Sheet 1. Related misconceptions about suprema continue to be a common source of errors on exams and problem sheets.
- Warning 1.53 has been expanded and moved earlier in its section. This continues to be a common source of errors on exams and problems sheets.
- Definition 1.15 on bounded subsets of metric spaces has been expanded to include a convenient equivalent definition for normed spaces. The proof of the equivalence, which is straightforward, is given in Exercise 1.1.11.
Changes in 2023/2024.
The most noticeable change this year is that I have rephrased several definitions to privilege subsets over subspaces. For instance, Definition 4.16 now defines compact subsets of metric spaces directly, without references to metric subspaces. While logically equivalent, it is hoped that this will be more intuitive for many students. Connections with metric subspaces are now made explicitly in Lemma 4.3, Lemma 4.8 and Lemma 4.19, which had previously been exercises. Corollary 1.27 and Lemma 1.28, which are useful tools for dealing with metric subspaces, have also been promoted from exercises to the main narrative.
Section 1.6 on higher-order functions has been significantly expanded to include a refresher on basic terminology for functions (Definition 1.52, Warning 1.53, Definition 1.54, Example 1.55) which often trip students up on exams. There is also a completely nonexaminable remark on higher-order functions in Python.
Other small changes include:
- There is a new footnote to Example 1.7 reminding students what it means for a real-valued function to be bounded.
- Solutions which have been assigned in problem sheet X will now have (PSX) added to their title rather than an asterisk.
- Exercise 2.1.3 has been slightly expanded.
- The html version now uses svg for MathJax by default. This avoids a long-standing bug in MathJax where ‘overlines’ (which we use to denote closures) do not render on all zoom levels.
- The table of contents on the side of the html version is now expandable/collapsible, although I am open to changing this.
Further changes made during the semester will be recorded in Appendix I.
Changes in 2022/2023.
Perhaps the most obvious change this year is the addition of new problem sheet, nominally ‘due’ in Week 1, on Analysis 1 facts about quantifiers, limits and suprema. While many students will find this routine, my hope is that it will be a useful diagnostic for those whose Analysis 1 skills are a bit rusty.
The remaining changes are not particularly dramatic, and mostly concern the exercises. Over the course of the previous semester many additional hints were added, and some of the longer exercises were split up into parts. Since then, I have added several new exercises:
- Exercise 1.1.5 introduces the standard \(\n\blank_1\) and \(\n\blank_\infty\) norms (on \(\R^2\) only). These are useful examples to have in mind, and the inequalities requested in the last part are used implicitly in much of the lecture notes. The follow-up Exercise 1.7.1 investigates whether these different normed spaces are isometric.
- A new exercise asks students to verify that some basic operations on normed an inner product spaces are continuous. These facts are used many times in the lecture notes, and are a good exercise in thinking about product spaces.
- Exercise 2.1.4 has been expanded to include a part about a subset not being dense.
- Exercise 5.2.4 on complex polynomials in two variables has been split off from the proof of Theorem 5.5.
Other changes include:
- Definition 1.57 has been snuck into Section 1.6. The standard notation for images and preimages has been used throughout the lecture notes; now, rather than being accompanied by a reminder of what this notation means, there is simply a link to this definition.
- I’ve put comments on the ‘examinability’ of Section 2.3, Section 2.4 and Section 6.2 from the start.
- This year I plan to summarise the discussion in example classes in a separate appendix rather than including this as part of the weekly schedule.
- PreTeXt has (finally!) added an option for figures to run on a separate counter. This will allow me to add additional figures during the semester without altering the numbering of important theorems and lemmas.
Changes in 2021/2022.
Probably the most noticeable change this year is the way the completions are constructed in Section 2.2. I have replaced the elegant and efficient argument using the completeness of \(C_\bdd(X)\) with a somewhat more involved argument using equivalence classes of Cauchy sequences. Both arguments are of course standard, but the hope is that the new one helps to build intuition for how to work in completions.
I have also added two short sections. Section 1.6 sets out clear notational conventions for mappings between spaces of functions. Section 3.3 very briefly recalls some basic properties of the Riemann integral which are used in Section 3.4, and also contains an important exercise which lays the groundwork for the proof in Lemma 3.11 that \(C^0([a,b])\) is not complete with the \(L^2(a,b)\) inner product.
I have added several additional remarks and ‘conventions’ explaining points of notation which commonly cause problems. While in previous years metric subspaces of \((X,d)\) were typically denoted \((Y,d_Y)\text{,}\) this year I have instead defaulted to \((Y,d')\text{.}\)
Finally, I have added a variety of new exercises, split some existing exercises into several parts, and removed some other exercises. Many of the new exercises involve testing definitions on simple concrete examples.
One change I would consider making in future years is to introduce so-called ‘diagonalisation’ arguments involving sequences of sequences, for instance in the proof of the Arzelà–Ascoli.
Changes in 2020/2021.
This set of lecture notes was originally written by Roger Moser, and has remained mostly unchanged over the last few years. This year I have added some new exercises and figures, and converted the document to PreTeXt in order to obtain both pdf and html output. I have also added a new section to Chapter 2, Section 2.4, on the completions of normed and inner product spaces. The major application is in Section 3.4, where \(L^2(a,b)\) is studied as the completion of an incomplete inner product space.
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