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Basic Analysis II: Introduction to Real Analysis, Volume II: Errata

Every book (no matter how much you paid for it) has errors and typos, especially text that is new in a given edition. I try to be as transparent as possible about any errors found, and I try to fix them as quickly as possible. Please do let me know if you find any errors or typos, so that they may be fixed. A free book relies on readers sending in corrections.

This page lists errors in the various editions. Nonmathematical typos, misspellings, and grammar or style problems are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions may suffer from recently discovered errata as well.

July 11th 2023 edition, Version 6.0 (edition 6, 0th update):

  1. Exercise 8.5.8 part e is incorrect, this map is not open. A small open rectangle near the origin will check that. Thanks to Nikita Borisov and Bob Strain for finding this erratum.
  2. In Exercise 10.1.4, the S and the R in the second sentence are reversed making the exercise make no sense. What the second sentence should say is "If \(f\) is a bounded function such that \(f(x) = 0\) for \(x \in S \setminus R,\) show that \(f \in {\mathcal{R}}(S)\) and \(\int_S f = 0.\)" Thanks to Bob Strain.

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

  1. The last line of the discussion before Proposition 8.1.11 says that \(0 \in Y\) when it should say \(0 \in \operatorname{span}(Y).\)
  2. The conclusion of Exercise 8.4.10 is not true with the given hypotheses. \(\frac{\partial h}{\partial x}\) also needs to exist and be continuous. Thanks to Jesse Wallace for bringing this up. This question will be moved to 9.1 in the next edition as that's the natural place for it.
  3. In Exercise 8.5.6, the interval \((c,d)\) could be infinite, that is, \(c\) and \(d\) need to be allowed to be \(\pm \infty,\) but this is not stated. Better just write \(I\) and call it an open interval instead.
  4. In the proof of Proposition 10.1.12, a priori the infima are taken over independent partitions, that is, infimum of \(U(P,f)-L(Q,f)\) is zero when taken over partitions \(P\) and \(Q .\) To get that infimum of \(U(P,f)-L(P,f)\) is zero, we must take a common refinement first.
  5. On page 102, in the displayed equation about halfway through, the \(R_k\) should be \(R_j .\)
  6. On page 104, in Example 10.3.6, the last displayed inequality starts with \(m^*([a,b]) \geq \) which should be removed. The inequality gives that \(m^*([a,b]) \geq b-a \) by taking the infimum.
  7. In Lemma 10.3.9, the domain of \(f\) should be \(U\) not \(B.\) While \(U=B\) would work if \(B\) is open, it doesn't work if \(B\) is closed as we have not defined differentiable functions of several variables on non-open sets.
  8. In the proof of Riemann-Lebesgue (Theorem 10.4.3), we claim that if \(R_j \subset T_\ell\), then \(M_j-m_j < 2\epsilon .\) That is actually correct but misleading, we have obtained above that \(M_j-m_j < \epsilon .\)
  9. In Exercise 10.5.3, the \(f(x)\) and \(g(x)\) are swapped in the integral.
  10. In Example 10.6.5, it says \(D_r := B(p,r)\) is a closed disc, when it is clearly not closed. Just remove the word "closed".
  11. In Exercise 11.2.5, obviously \(X\) cannot be an arbitrary compact metric space, it was meant to be the compact interval \([a,b].\)
  12. In Proposition 11.2.7, the hypothesis that \(Y\) is complete is missing, even though it is used in the proof. (Thanks to Judah Nouriyelian)
  13. In the proof of Theorem 11.7.1, in the footnote about the delta function, the \(g(x)\) under the integral sign should be \(g(t) .\)
  14. In Exercise 11.7.10, the hypothesis that \(\mathcal{A}\) vanishes at no other point of \(X\) is missing. Thanks to Jessee Wallace for bringing this up.
  15. When talking about writing \(s_N(f;x)\) as a convolution, there is no need to change \(x-t\) to \(t-x,\) that's a typo.
  16. In Exercise 11.8.9, the boundary value should obviously be \(f(e^{i\theta}) = \sum_{n=0}^\infty c_n e^{in\theta}.\) The \(n\) was missing in the exponent on the right hand side.

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

  1. On page 10, we didn't define span for empty set, take \(\operatorname{span}(\emptyset) := \{ 0 \} .\) Similarly best to explicitly mention that an empty set is (by convention) linearly independent and spans \(\{ 0 \}.\)
  2. The example starting on page 20 and continuing on page 21: differentiation on continuously differentiable functions of course returns only a continuous function, not a differentiable function, so that should be mentioned.
  3. On page 24, Proposition 8.2.7 is only stated for functions whose codomain is \(L({\mathbb{R}}^n,{\mathbb{R}}^m)\) but we also need it for the domain in just the next subsection (showing that determinant is continuous). The fix is to show the inequality that the euclidean norm of the entries is bounded by square root of \(n\) times the operator norm, to show that the metrics are equivalent. Takes about the same space on the page and proves more.
  4. On page 27, we should be supposing without loss of generality that \(\gamma_1 \not= 0\) (and not 1 as it says).
  5. On page 28 when defining the the first type of elementary matrix, it says \(k\)th row, when it should say \(j\)th row. Thanks to Wang KP for spotting this.
  6. On page 33, proof of Proposition 8.3.2. The letter \(x\) appears as the point at which we are differentiating and then in the last three lines of the proof it changes to be an arbitrary unit vector, so that needs to be changed to a different letter.
  7. Exercise 10.2.5, the function \(h\) must is defined on \([c,d]\) not \([a,b].\)
  8. In the proof of Lemma 10.3.9, we use Proposition 8.4.2 for a closed set, though that was not proved (it still works). But it is easier to prove the lemma for open \(B\) first anyway and just use the closure.
  9. In the proof of Proposition 10.3.10 towards the end, the balls \(B_1,\ldots,B_k\) must be chosen to be of radius at most \(\delta/2\) and intersect \(E\) for them to be definitely in \(U.\)
  10. In the same proof, the final estimate, there is a parenthesis missing around the \(Mr_j\) as that whole number must be raised to the \(n\)th power leading to \(M^n \epsilon .\)
  11. In Exercise 10.3.7, it should say "Show that \(G\) is of measure zero."
  12. In the proof of Theorem 10.4.3, the bound \(|f(x)| \leq B\) for all \(x\) implies that \(M_j-m_j \leq 2B\) (the intequality was strict by mistake).
  13. The last item (iv) of Corollary 10.4.4, and hence Exercise 10.4.8: First \(m=n,\) and second we need to assume \(g(R) \subset R'.\)
  14. In Exercise 10.5.7, the Jordan measurable sets should be closed.
  15. At the end of the proof of 11.8.12, in first line of the last estimate, the \(g\) should always be \(\bar{g}.\) Thanks to Wang KP for spotting this.
  16. On page 181, when we write as a remark that the \(D_N(x)\) are partial sums of the delta functions, it should be \(2\pi \, \delta(x),\) the \(2 \pi\) is missing.

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

  1. In the proof of Proposition 9.3.3 on the lower half of page 78, starting with "We wish to take the limit," all the \(\omega\) up to but no tincluding the last one in that paragraph should be \(\omega_j.\) Thanks to Wang KP for spotting this.
  2. In the first sentence of the proof of Proposition 10.3.2, "then \(R\) is contained in a closed ball of ...", that should be \(C,\) not \(R.\) Thanks to Wang KP for spotting this.

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

  1. Definition 9.2.1, page 66: The definition of "a simple path" is still not quite right for non-closed simple paths. We should require that if \(\gamma\) is not one-to-one on the entire interval then it should be equal at the endpoints (no biting itself in the middle): "A path \(\gamma \colon [a,b] \to {\mathbb{R}}^n\) is simple if either 1) \(\gamma\) is one-to-one, or 2) \(\gamma|_{[a,b)}\) is one-to-one and \(\gamma(a)=\gamma(b).\)" The wording of Exercise 9.3.7 already assumed this correct definition.
  2. In Definition 9.2.5, page 68, for a piecewise smooth \(h\) the fact that all one-sided limits of \(h'\) are nonzero is forgotten, although it is correctly explained below, and correct in the definition for a smooth reparametrization.
  3. Exercise 9.2.14, page 76: The explanation of what not being one-to-one except on a finite set was misleading it was only a consequence not an if and only if (to be an if and only if it was missing that the preimages are always finite). It is better to just state directly what being one-to-one except on a finite set means. Same issue is on page 71 where the exercise is referred.
  4. Exercise 10.5.3, page 115: The function defined on \(U\) should clearly be named something else than \(f.\) Say \(\varphi.\)
  5. Exercise 10.5.4, page 116: Definition of \(W\) should subtract the complement of \(U,\) not \(U.\) Actually an easier way to write what \(W\) should be is \(\bigl( U \times (0,2) \bigr) \cup \bigl( (0,1) \times (1,2) \bigr).\)
  6. Exercise 10.6.4, page 120: In part c), the first \(dy\) should be a \(dx,\) that is, the integral should be \(\displaystyle \int_{\partial B(p,r)} \frac{-y}{2} ~ dx + \frac{x}{2} ~ dy.\)
  7. In the proof of 10.7.2, throughout the proof, \((1+4\sqrt{n}\, \epsilon)\) should be \({(1+4\sqrt{n}\, \epsilon)}^n\) and in the long estimate on page 123, the three instances of \((1+4\sqrt{n}\, \epsilon)\) should be \(\frac{1}{{(1+4\sqrt{n}\, \epsilon)}^n}.\)
  8. The statement of Theorem 11.3.1 is confusingly worded. The three items are three distinct possibilities.
  9. In Lemmas 11.5.1 and 11.5.4 and Theorem 11.5.5 (FTA), the hypothesis that the polynomials are nonconstant is missing.
  10. In the caption to Figure 11.7 it says the vertical axis is from \(-e^6\) to \(e^6,\) but it should say \(-e^4\) to \(e^4.\) The approximate value \(54.6\) is correct.

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

  1. On page 35, towards the end of the proof of the chain rule, it says that \({\|f(p+h)-f(p)-Ah\|}{\|h\|}\) is bounded, when it should be, of course, \(\frac{\|f(p+h)-f(p)-Ah\|}{\|h\|}.\)
  2. On bottom of page 47, defining \(\varphi_y,\) the domain of \(\varphi_y\) should be \(V\) not \(C.\)
  3. Definition 9.2.1, page 67: definition of "a simple path" is not quite right, it should be that \(\gamma\) is one-to-one on \([a,b),\) not just \((a,b).\) The error is then carried over in the couple of places where "simple" is used in the examples and exercises: Example 9.2.2, discussion on page 71, exercise 9.2.13.
  4. On page 145, end of proof of Theorem 11.3.9. We have only proved that the set of cluster points of \(E\) is open (we claimed that we proved that \(E\) is open), but the set of cluster points is also closed and so equal to \(U\) anyway.
  5. On page 157, when we say we enumerate \(X\) in the proof of Proposition 11.6.5, it makes it seem like \(X\) is necessarily infinite, which is the important case, but the proof should work even for finite \(X.\)
  6. On page 160, Exercise 11.6.1 only asks about why the sequence does not converge uniformly, it should ask about a subsequence. The solution is essentially the same, and we are asking about why ArzelĂ -Ascoli does not apply so we must ask about a subsequence.

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

  1. On page 102, middle of the page in the estimate for \(\sum r_k^n,\) the right hand side should be \({(4\sqrt{n})}^n \epsilon.\)

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

  1. On top of page 35 (right after Theorem 8.3.7, Chain Rule) it says \((f \circ g)'\) when it ought to be \((g \circ f)'.\) Thanks to Trevor Fancher.

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

  1. On page 10, when solving for \(x_j,\) the coefficients must be negative, that is, \( x_j = \frac{-a_1}{a_j} x_1 + \cdots + \frac{-a_{j-1}}{a_j} x_{j-1} + \frac{-a_{j+1}}{a_j} x_{j+1} + \cdots + \frac{-a_k}{a_k} x_k . \)
  2. In Exercise 8.2.13, part a), it should say "for all \(c\in {\mathbb{R}}^n\)" instead of just "for"
  3. On page 76, in the definition of the Darboux sums, \(P\) is a partition of \(R.\)
  4. In Exercise 8.3.8, the hypothesis should say \(\nabla f(0,0) = (0,1).\) Thanks to Trevor Fancher.
  5. In the Inverse function theorem, Theorem 8.5.1, the \(U\) should be an open set.
  6. In Exercise 8.5.6, the hypothesis should be that the first component of \(\nabla f(t)\) is not zero.
  7. In the proof of Theorem 9.1.1, an offhand remark is made about replacing \(h\) with \(\frac{1}{n}.\) That wouldn't work, we need an arbitrary sequence \(\{ h_n \}\) converging to zero.
  8. Exercise 9.1.7 part c) is impossible (not true). Removing part c makes the exercise quite uninteresting so it will be replaced with a different exercise (demonstrating the same issue) in the next edition.
  9. In definition of smooth path, we really assume that derivative of a function of one variable has been defined, although this was only properly defined for the components. Thanks to Trevor Fancher for pointing this out.
  10. In the proof of Proposition 9.2.6, the restriction of \(h\) in the first paragraph is to \([r_{j-1},r_j],\) not \([t_{j-1},t_j].\) In the second paragraph, the \(\varphi\) should be a \(\gamma.\)
  11. In Definition 9.2.7, the functions \(f_1,\ldots,f_n\) should be named \(\omega_1,\ldots,\omega_n\) as those are used in the definition.
  12. In Definition 9.2.9, when talking about the partition being minimal we mean \(t_1,t_2,\ldots,t_{k-1}\) not \(t_2,t_3,\ldots,t_{k-1}.\) Also in the definition, in the displayed equation for the piecewise smooth definition, the last interval is marked as \([t_{n-1},t_n]\) when it should of course be \([t_{k-1},t_k].\)
  13. Exercise 9.3.6 should require that \(U_1 \cap U_2\) is path connected, we don't want to have to use another exercise here.
  14. Exercise 9.3.9 is for an open set \(U,\) while this is tacitly implied as we have not defined path-connected for other sets, it should be explicit.
  15. In proof of Theorem 10.1.15, the expression \(M_i-m_i\) should be \(M_k-m_k .\)
  16. In the proof of Proposition 10.3.2, the proof of the reverse direction does not make sense. There is no need estimate \(r_j^n\) with \(r_j.\)
  17. In the construction of Cantor set in Example 10.3.8, \(j\) and \(n\) are used interchangeably. It should all just be \(n.\)
  18. In the proof of Claim 1 of the proof of Stone-Weierstrass 11.7.12, \(N\) and \(n\) are both used for the same thing, it should be one or the other.

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