MA30252: Advanced Real Analysis

Add a link to today’s Re:View recording to the front matter of the notes.

Fix a typo in Exercise 1.1.3: in several places single bars \(\abs\blank\) were used for standard Euclidean norm \(\n\blank\) on \(\R^2\text{.}\)

Flip the sign of the function \(f\) in the second part of Exercise 1.1.8 for obscure reasons.

Expand the first and especially the second hint to Exercise 1.1.9.

Tweak the wording of Exercise 1.1.5 and Example 1.7. The mathematical content of these exercises is unchanged.

Add a link to Re:View recording to the start of Section 1.1. While I will continue linking to recordings throughout the term, I won’t be making any further changelog entries about this.

Post Problem Sheet 2. While I will continue releasing new problem sheets through the term, I won’t be making any further changelog entries about this.

Exercise 1.1.5 (PS2) Some norms on \(\R^2\).

Exercise 1.1.5.

Improve the hints for the second part of Exercise 1.2.2.

Fix a typo in Exercise 1.3.1: \(y_0 \in Y\) was written when it should have been \(y_0 \in X\text{.}\)

Split the proof of Theorem 1.40 up into steps, and simplify one of these by citing Exercise 1.1.11 and Exercise 1.3.3.

Over the summer I had expanded Definition 1.15 on bounded subsets of metric spaces to include an equivalent (and quite useful) definition for normed spaces, but forgot to mention this in class. I will do so today. I had also forgotten to record this change in the list of Changes in 2024/2025.

Some small teaks to the Weekly Schedule. Having finished Section 1.3 on Friday, we are now slightly ahead of the schedule from last year, which is a good place to be. There is also a possibility that I will be away during Week 5.

Fix a typo in Definition 1.54: \(g=f|_A\) was written when it should have been \(f=g|_A\text{.}\)

Very small tweak to the hint for Exercise 1.4.2.

Add an extra sentence to Lemma 1.4, and two extra bullet points to Convention 1.16, to further clarify our conventions for using norms induced by inner products and metrics induced by norms.

In response to a request on the Week 4 Questionnaire, add the short Section 1.8 to the end of Chapter 1. This is an attempt to list all of the facts about cardinality (countable vs. uncountable) that you might ever want to use in this unit.

Make some updates to Appendix A.

Expand Remark 3.4.

Add a few sentences to the start of Section 1.8.

Add exam-related question to Section H.2.

Add another hint to Exercise 2.1.4.

Release solutions to the unassigned Exercise 1.1.7.

Fix a typo in the hint to the last part of Exercise 4.3.5.

Remove “and inner products” from the title of Exercise 3.4.2, since this exercise does not, in fact, ask you to estimate any inner products.

Tweak Appendix A.

Add “and Part b” to the end of the second hint for the last part of Exercise 3.3.1.

Expand the hints to Exercise 3.2.5.

Add an additional hint to Exercise 4.1.1, really more of a reminder to answer the question as stated.

Tweak the statement of the first part of Exercise 4.3.5 to emphasise that when talking about \(F\) being bounded we are thinking of it as a subset of \(C^0([0,\pi])\text{.}\)

Add a hint to the last part of Exercise 4.3.5.

Post Problem Sheet 8 a bit late!

Add Lemma 1.72 (and an example) to Section 1.8 on cardinality, as this is useful for the last part of Exercise 5.1.1.

Add “where \(a_n\) and \(b_n\) are not both zero” to the end of Definition 5.4, as otherwise \(p\) should technically be called a “trigonometric polynomial of degree \(\le n\)”. In the context of this unit the distinction is not particularly important.

Change notation from \(\mathrm{PL}(a,b)\) to \(\mathrm{PL}([a,b])\) in Exercise 5.2.3 to better match the notation in Chapter 3.

Slightly tweak the wording of the hints to the first and last parts of Exercise 4.5.2, and add additional hint/warning to the second part of Exercise 4.5.2.

Slightly tweak the wording of Exercise 5.1.1 and completely rework the (relatively big) hint to the first part to recommend an argument using sequences.

Rephrase the hints to the last part of Exercise 5.1.1 to refer to the relevant results in Section 1.8.

Add a link to the formula sheet in Subsection H.1.2.

Copy some student questions from last year about the exam into Section H.2.

Add commentary to Appendix H about the five past exams available on the Library website.

Replace the ‘binomial coefficient’ \(\binom{\frac 12}{n}\) in the proof of Lemma 5.16 with its definition.

By popular request, add a proof of Proposition 5.15 on the convergence of a particular Taylor series needed in Section 5.2 by adding Exercise 5.2.6, Exercise 5.2.7 and Exercise 5.2.8 as unassigned exercises and releasing their solutions. I should stress that these exercises are not relevant for the exam. I inherited them from a previous lecturer for the unit, and have not checked them carefully.

Correct the problem class summary in Example 9, which said ‘Lipschitz’ in one place when it should have \(C^1\text{.}\)

Update the Weekly Schedule to show our plans for Revision Week.

Add Question H.13 to Section H.2.

Add text to the start of the unassigned Exercise 5.2.6, Exercise 5.2.7 and Exercise 5.2.8 emphasising that these are not especially useful for exam preparation. See the above changelog entry from 4 December for more details.

Add links to the first two revision lectures to Appendix C.

Add commentary to the section on The 2020 exam explaining why the definition of “totally bounded” in the official solution isn’t exactly Definition 4.7.

Add more student questions to Section H.2.

Fix small typo in the solution to the first part of Exercise 2.1.3: \(\delta\) was written instead of \(\varepsilon\) somewhere.

Add another student question to Section H.2.

Add commentary to the section on The 2021 exam about a typo in the solution to Question 3(c)(ii).