# Basic Analysis II: Introduction to Real Analysis, Volume II: Changes

I preserve numbering as much as possible. So for example exercises are only added with new numbers, so that old exercises are not renumbered, etc. I try to preserve pagination as well though adding a page in the middle is sometimes unavoidable.

May 16th 2022 edition, Version 2.6 (edition 2, 6th update):

The list of changes may seem long, but vast majority are rather minor changes. The focus was purely on fixing issues and improving clarity, not on any new material nor restructuring. No numbers were changed. A couple of exercises were modified a bit to fix errata, and 3 exercises were added to 10.3 to address part of the exposition that was perhaps a bit too quick in 10.5 and could have really been done in exercises. Pagination has changed very slightly in a few places.

1. The word "symmetric" as used for a dot product is in the index.
2. Define properly that the span of the empty set is $$\{0\}.$$
3. In Proposition 8.1.11, add a sentence about $$0$$ being in $$Y$$ before the proposition.
4. After Definition 8.1.12 note explicitly that no vector in a linearly independent set can be zero, that $$\{0\}$$ is the only vector space of dimension 0, and note that the empty set is trivially linearly independent and spans $$\{0\}$$.
5. Improve the proof of part (ii) of Proposition 8.1.14. And in the beginning mention that $$d=0$$ is trivial for all statements.
6. When stating that $$L(X,Y)$$ is a vector space, explicitly mention that 0 is the linear map taking everything to 0.
7. In Proof of 8.1.18, flip the $$=0$$ to be at the front as that is slightly clearer.
8. Add a short paragraph with the example of $$X={\mathbb{R}}^1$$ for the operator norm right after the definition (8.2.3)
9. Add a footnote about the acronym GL for general linear group to Proposition 8.2.6.
10. In the proof of 8.2.6 part (ii), note that (i) immediately implies that $$GL(X)$$ is open, meaning we only need to prove that the inverse is continuous. In the same proof, note the use of the rank-nullity theorem.
11. When giving what the column of the matrix represents be more specific to say that the $$j$$th column represents the $$A x_j$$ vector.
12. Reorder the discussion about matrix multiplication on page 24, and explicitly note that unless otherwise specified we identify the set of m-by-n matrices and $$L({\mathbb{R}}^n,{\mathbb{R}}^m).$$
13. To fix an erratum, Proposition 8.2.7 was changed. In the discussion before hand, show that the euclidean norm of the entries is bounded by square root of $$n$$ times the operator norm. Then the proposition is about the topology being the same. We also add the conclusion about continuous functions both with the domain and codomain being the operators (or matrices).
14. Exercise 8.2.4, part b). Give a hint on how one could describe $$B(0,1)$$ as for example a convex hull and tell the student to think about it in $${\mathbb{R}}^2$$ and $${\mathbb{R}}^3.$$ It is still somewhat open to interpretation, but in some sense, this part is meant to make the student think, so that's OK.
15. Exercise 8.2.15: Change the hint, I cannot follow the previous hint, I wonder if it is a typo.
16. Exercise 8.2.16: Add a hint to part a (it is still possibly quite challenging). Also emphasize that we are working on $${\mathbb{R}}^n$$ with the euclidean norm.
17. Before Theorem 8.3.7, when giving the intuition, use $${\mathbb{R}}^n,$$ $${\mathbb{R}}^m,$$ and $${\mathbb{R}}^k$$ instead of $$X,$$ $$Y,$$ and $$Z.$$
18. Add "tangent vector" to the index and mark it in 8.3.3 when defining derivative of a curve. Also mention it in the caption to Figure 8.5, along with mentioning that we are assuming $$\gamma$$ is defined up to the endpoints for clarity.
19. After definition of directional derivative, do mention that we could have used any path $$\gamma$$ as long as it had the right derivative at zero.
20. In proof of Theorem 8.5.1, mention what $$g$$ is in the proof. Then during the proof that $$W$$ is open, use $$x_1$$ for a random point in $$C(x_0,r)$$ to avoid confusion with the fixed point $$x$$ later in the paragraph.
21. In Example 8.5.4, note that the $$f$$ is onto $${\mathbb{R}}^2 \setminus \{ (0,0) \} .$$
22. In Theorem 8.5.8, write $$A=[A_x~A_y]$$ in the statement as $$A$$ is used towards the end of the proof just as it is used as in the previous proposition.
23. In proof of Theorem 9.1.1, emphasize that the $$y_1$$ may have depended on $$x,$$ even though the estimate does not.
24. Reorder the first two pages of 10.1 a little bit by moving the remarks out of the definition. It flows a bit better now.
25. In Definition 10.3.1, explicitly mention that we are allowing $$\infty .$$
26. In proof of Proposition 10.3.2, we explicitly mention that the rectangles in question are all of positive volume, and there is no reason to consider closed rectangles, so don't mention that.
27. In Exercise 10.3.6, in the hint, use the norm rather than an abstract metric notation, we are doing this in the euclidean space.
28. Add Exercises 10.3.14, 10.3.15, 10.3.16. 10.3.16 is needed in the proof of Proposition 10.5.3 and the other two exercises can make it a bit easier to prove.
29. After the definition of oscillation in 10.4, note that it could be infinite if the function is not bounded.
30. In subsection 10.4.1, use $$D$$ instead of $$S$$ for the domain of the functions. We wanted a random set, but then in the next subsection $$S$$ is used for the set of singularities so this could be confusing.
31. In the proof of 10.4.3, the estimate for $$T_x$$ works with $$\epsilon$$ rather than $$2\epsilon$$ as the $$T_x$$ can be put into a $$\delta$$ ball given $$o(f,x,\delta) < \epsilon,$$ so mention all that. The final estimate on page 110 is adjusted.
32. In the same proof mention explicitly that the volume of the second set of subrectangles is bounded by the sum of the volumes of $$O_j$$ and that's why it's bounded by epsilon (and that's strict).
33. In the final estimate on page 110 we put strict inequalities since that's what we have (though this is not a big deal).
34. On page 111, explain a bit more explicitly why $$M_j-m_j \geq \frac{1}{k}$$ when $$R_j \cap S_{1/k}$$ is nonempty.
35. In section 10.5, add a note that multiplying two functions that are Riemann integrable is still Riemann integrable.
36. Exercise 10.5.7: The Jordan measurable sets are assumed closed (this was an erratum).
37. In Proposition 10.5.6, Theorem 10.7.2, Exercise 10.7.2, Exercise 10.7.6, the words "closed bounded" is perhaps confusing because it could be interpreted as closed in the subspace topology. Replace with "compact".
38. In Exercise 10.7.2 and 10.7.4, don't specify "closed bounded" rectangle, all rectangles as we defined them are bounded.
39. In the first sentence of section 11.2 reference chapter 6 to explicitly mention what we are generalizing.
40. Break out the definition of "uniformly Cauchy" to give it explicitly before Proposition 11.2.3, and restate Proposition 11.2.3 to make the "Cauchy-complete" only be a hypothesis in the direction that is needed: i.e., Cauchy implies convergent.
41. After the $$M$$-test, explicitly note that it also proves absolute convergence.
42. Simplify and clarify the wording of Examples 11.2.5 and 11.2.6.
43. Simplify the statements of Proposition 11.2.7, and Corollary 11.2.8 and don't require Cauchy-complete here since it is not required.
44. In the proof of Proposition 11.2.7, do not just say "by continuity of the metric" as that is not so clear what is the estimate. Give the explicit estimate that we are taking a limit of.
45. In the proof of Example 11.2.15, write down the expression for the difference quotient right after the definition of $$\delta_m$$ so that the reader knows where $$\gamma_n$$ is coming from.
46. In the proof of 11.7.1, when mentioning the delta function give its integral property in the footnote.
47. At end of subsection 11.8.4, expand a little bit on the delta function since we are putting it into a displayed equation.
48. In the last estimate of the proof of 11.8.12, there is an unnecessary step that was there to use a real version of Cauchy-Schwarz inequality (but the complex one is referred to here). So the middle line in the estimate can just be removed.
49. Fix the errata from the last revision.

November 9th 2021 edition, Version 2.5 (edition 2, 5th update):

1. Make line in Figure 9.6 a bit bolder to make it easier to pick out.
2. In Proposition 10.3.7, use $\ell$ for the number of balls to make it clear that the number is quite likely different from the number of rectangles.
3. Fix a couple of uses of "=" where ":=" is more appropriate.
4. Some minor clarifications and fixes to style and grammar.
5. Fix the errata from the last revision.

June 8th 2021 edition, Version 2.4 (edition 2, 4th update):

1. On page 11, add a short note about the $d+1$ linearly independent vectors from the definition of dimension.
2. In Exercise 8.4.7, assume $q$ is not identically zero. The result is vacuously true even if $q$ is identically zero, but there is no reason to make students think about this rather stupid technicality.
3. The proof in Example 8.1.25 is hopefully clearer.
4. In Definition 8.3.8, remove the definition of the notation $D_j f.$ We never used it later.
5. In Definition 9.2.1, change the definition of "simple" for non-closed paths. Typically a path that bites itself back in the middle is not called simple, so rule out that case.
6. Rename Subsection 10.3.3 to "Images of null sets under differentiable functions".
7. In proof of Lemma 10.3.9, say how to prove it for open balls.
8. Improve Definition 10.6.1 to be (much) simpler (though equivalent): Only take a finite disjoint union of simple closed sets as we are assuming $U$ is bounded anyway.
9. Reorder the proof of Theorem 10.7.2 a little bit to make it more logical.
10. On page 148, add a short note that the $e^{z+w}=e^ze^w$ leads to a quick computation of the power series at any point.
11. In Corollary 11.3.7, emphasize that that $a$ is any complex number, since just above it was a real number.
12. In definition of the exponential on page 147 (the definition of $E(z)$) explicitly say that this means that it is analytic.
13. Be more precise in Exercise 11.4.9 to say to derive the power series at the origin.
14. In Example 11.6.3, emphasize that $f_n$ are continuous.
15. In 11.8.2, when saying we could develop everything with sines and cosines, give the actual form of the series and refer to Euler's formula, so that when we later call such series also Fourier series, the reader is not confused.
16. Fix the errata from the last revision.

June 10th 2020 edition, Version 2.3 (edition 2, 3rd update):

1. Improve the introduction.
2. In the statement of inverse function theorem, remove the definition of $q,$ it is not used.
3. In proof of second part of 10.4.3 (Riemann-Lebesgue) use $S_{1/k}$ rather than $S_k$ for consistency with the first part of the proof.
4. In the path integral chapter, consistently use the words "path integral" rather than sometimes "path integral" sometimes "line integral".
5. Mention that the nowhere differentiable function construction that we gave is due to Takagi (although it is the one given in Rudin, not the form of it usually attributed to Takagi).
6. Greatly simplify the function in Example 11.6.4.
7. In Exercise 11.6.1, the question should ask about there being no subsequence that converges uniformly just like 11.6.2. The question is about why Arzelà-Ascoli fails.
8. Many minor language and style improvements as well as some minor clarifications.
9. Fix the errata from the last revision.

May 15th 2019 edition, Version 2.2 (edition 2, 2nd update):

1. Several minor grammar and style fixes.
2. Fix the errata from the last revision.

October 11th 2018 edition, Version 2.1 (edition 2, 1st update):

1. Links are now https.
2. Fix errata.

May 7th 2018 edition, Version 2.0 (edition 2, 0th update):

Numbering of definitions, examples, propositions changed in 8.1, 8.3, 10.1. Numbering of exercises is unchanged, except for 9.1.7 which was replaced due to erratum.

1. New Section 10.7 on change of variables.
2. New Chapter 11 on Arzela-Ascoli, Stone-Weierstrass, power series, and Fourier series.
3. A List of Notations is added at the back as in volume I.
4. In the PDF the pages have been made slightly longer so that we can lower the page count to save some paper.
5. Add figure showing vector as an arrow and discussion about this for those that do not remember it from vector calculus.
6. Add a paragraph about simple algebraic facts such that $0v=v.$
7. Add footnote about linear independence for arbitrary sets in 8.1.
8. Add example that span of $t^n$ is ${\mathbb{R}}[t].$
9. Add remark about proving a set is a subspace.
10. We use the words "linear operator" for $L(X,Y),$ and it is for $L(X)$ that we say "linear operator on $X$", so update the definition appropriately.
11. Add convexity of $B(x,r)$ as a proposition since we use it so often.
12. Add Exercise 8.1.19.
13. Proposition 8.2.4 doesn't need $Y$ to be finite dimensional, same in the Exercise 8.2.12, so no need to assume it.
14. In Proposition 8.2.5, emphasize where the finiteness of dimension is needed.
15. Use $GL(X)$ as notation for invertible linear operators.
16. Give more detail on why mapping between matrices and linear operators is one to one once a basis is fixed.
17. Add a commutative diagram to the independence on basis discussion.
18. Reorder the definition of sign of a permutation to be more logical.
19. Add short example of permutation as transpositions.
20. Add Exercises 8.2.14, 8.2.15, 8.2.16, 8.2.17, 8.2.18, 8.2.19.
21. Add figure to Definitions 8.3.1 and 8.3.8.
22. Add Proposition 8.3.6, which was conspicuously missing.
23. Add figure for differentiable curve and its derivative.
24. Add figure to Exercises 8.3.5 and 8.3.6.
25. Add Exercise 8.3.14.
26. Add graph to figure in Example 8.4.3 (and adjust the formulas).
27. As application of continuous partials imply $C^1.$
28. Add Exercises 8.4.7, 8.4.8, 8.4.9, 8.4.10.
29. Fix up statement of the inverse function theorem in 8.4.
30. Add a couple of figures to proof of the inverse function theorem.
31. Add a figure to the implicit function theorem.
32. Add a short paragraph about the famous Jacobian conjecture.
33. Make the remark at the end of 8.5 into an actual "remark".
34. Add observation about solving a bunch of equations not just for $s=0$ for the implicit function theorem.
35. Add Exercises 8.5.9, 8.5.10, 8.5.11.
36. Add figure to 8.6.
37. In 8.6 cleanup the argument in the proposition and use only positive $s$ and $t$ for simplicity.
38. Add Exercises 8.6.5, 8.6.6, 8.6.7.
39. Refer to the new Proposition 7.5.12 about the continuity in 9.1.
40. Add figure to example in 9.1.
41. Exercise 9.1.7 replaced due to erratum. The replacement shows the same issue that the previous wrong exercise tried to.
42. Add Exercise 9.1.8.
43. Reorder the introduction of 9.2 a bit, and fix an erratum in that derivative at the endpoints was not really defined for mappings.
44. Add figure to Examples 9.2.2, 9.2.3, 9.2.11, 9.2.13, 9.2.18.
45. Add figure for definition of a function against arc-length measure.
46. Add figure to proof of path independence implies antiderivative in 9.3.
47. Add figure to proof that integral over closed paths being zero means that the integral is path independent in 9.3.
48. Add figure to Definition 9.3.5.
49. Change hint to Exercise 9.3.8.
50. Add Example 10.1.16 of compact support with a figure, following examples/propositions in 10.1 are renumbered.
51. Explicitly mention monotonicity of outer measures right after the definition (it is a rather easy exercise), and also allowing finite sequences of rectangles in the definition (a new exercise).
52. Add figure to definition of outer measure.
53. Clean up proof of Proposition 10.3.2.
54. Add Exercises 10.3.11, 10.3.12 (and a figure), 10.3.13.
55. Add corollary for the Riemann integrability theorem showing that it is an algebra, that min and max of two functions are Riemann integrable and so is the absolute value.
56. Add Exercises 10.4.6, 10.4.7, 10.4.8, 10.4.9, 10.4.10, 10.4.11.
57. Add Exercises 10.5.5, 10.5.6, 10.5.7.
58. In 10.6, add figure for positive orientation and a figure illustrating the three types of domains.
59. Many minor improvements in style and clarity, plus several small new example throughout.
60. Fix errata.

March 21st 2017 edition, Version 1.0 (edition 1, 0th update):

First version.