Analysis II

Third edition (hardcover)

108 corrections · 1 awaiting a page number

1. p. 18The proof of Lemma 1.4.3 should refer to Exercise 1.4.1, not Exercise 1.4.3.
2. p. 21In Exercise 1.4.8(c), add “(This part of the exercise requires the axiom of choice.)”.
3. p. 22In Definition 1.5.3, add “We call $(X,d)$ bounded if $X$ is bounded”.
4. p. 32In Corollary 2.2.3(b), “$f/g: X \rightarrow \mathbb{R}$ is also continuous at $x_{0}$” should be “$f/g: X \rightarrow \mathbb{R}$ is also continuous”, and similarly for “$cf: X \rightarrow \mathbb{R}$ are also continuous at $x_0$“.
5. p. 42In Exercise 2.5.5, $V \subset X$ should be $V \subseteq X$. In Exercise 2.5.13, “topological space” should be “Hausdorff topological space”. In Exercise 2.5.4, after “the trivial topology is not Hausdorff” add “if the space contains at least two elements”.
6. p. 43In Exercise 2.5.6, insert “and” between “if $x \in X$” and “$(V_n)_{n=1}^\infty$ is any…”.
7. p. 48In Exercise 3.1.5, $\lim_{x \in x_0; x \in E}$ should be $\lim_{x \to x_0; x \in E}$.
8. p. 71In Corollaries 3.8.18 and 3.8.19, “supported on $[0,1]$” is redundant and may be deleted.
9. p. 98100: Exercises 4.6.10-4.6.13 may be deleted, and the paragraph after Lemma 4.6.13 may be replaced with “Observe that with our choice of definitions, the space ${\bf C}$ of complex numbers is identical (as a metric space) to the Euclidean plane ${\bf R}^2$, since the complex distance between two complex numbers $(a,b), (a',b')$ is exactly the same as the Euclidean distance $\sqrt{(a-a')^2+(b-b')^2}$ between these points. Thus, every metric property that ${\bf R}^2$ satisfies is also obeyed by ${\bf C}$; for instance, ${\bf C}$ is complete and connected, but not compact.”
10. p. 122In Remark 5.5.2, “continuously differentiable” may be relaxed to just “differentiable”, and “twice continuously differentiable” may be relaxed to “continuously differentiable”.
11. p. 123In the sixth line, “Corollary 5” should be “Corollary 5.3.6”.
12. p. 140In the last paragraph, “continuous on $F$” should be “continuous at $x_0$“.
13. p. 146The definition of continuous differentiability needs to be detached from Definition 6.5.1 and placed near the beginning of this page.
14. p. 159In the first line of the third display in the proof of Theorem 6.8.1, the $f$‘s should be $F$‘s.
15. p. 161“if some other derivative $\frac{\partial f}{\partial x_j}$ is zero” should be “if some other derivative $\frac{\partial f}{\partial x_j}$ is non-zero”.
16. p. 176In Proposition 7.3.3, $\sum_{q \in J} q+E$ should be $\bigcup_{q \in J} q+E$ (two occurrences).
17. p. 178In Corollary 7.4.7, $m(B \backslash A) = m(B) - m(A)$ should be $m(B \backslash A) + m(A) = m(B)$.
18. p. 10In Exercise 1.1.8, a right parenthesis is missing at the end of the last sentence. In Exercise 1.1.11, ${\mathbf R}^n$ should be $X$.
19. p. 15In Proposition 1.2.15, $\supset$ should be $\supseteq$ (two occurrences).
20. p. 16In the first paragraph, the first parenthetical comment should be closed after “… and hence outside of $E$.” In the second parenthetical comment, the period should be outside the parenthesis. “The point 0” should be “The point $(0,0)$” (two occurrences).
21. p. 21In Exercie 1.4.7 (b), ${\bf R}^+$ should be $[0,+\infty)$.
22. p. 22In Definition 1.5.3, insert “for every $x \in X$” before “there exists a ball” (in order to keep the empty metric space bounded). Also, add the requirement that $r$ be finite.
23. p. 23In Theorem 1.5.8, $I$ should be $A$ in the statement of the theorem (four occurrences). In Case 2 of the proof, $B(y,r_0/2) \in V_\alpha$ should be $B(y,r_0/2) \subset V_\alpha$. One should in fact split into three cases, $r_0=0$, $0 < r_0 < \infty$, and $r_0=\infty$. For the last case, write “For this case we argue as in Case 2, but replacing the role of $r_0/2$ by (say) $1$“. In the proof of Theorem 1.5.8, $Y \subset \bigcup_{\alpha \in F} V_\alpha$ should be $Y \subseteq \bigcup_{\alpha \in F} V_\alpha$.
24. p. 26In Exercise 1.5.10, $n$ should be a natural number rather than a positive integer (in order to ensure that the empty set is totally bounded).Page 29: In Theorem 2.1.4(c), all occurrences of $\subset$ should be $\subseteq$.
25. p. 30In Exercise 2.1.7, $E \subset Y$ should be $E \subseteq Y$.
26. p. 33Add an additional Exercise 2.2.12 after Exercise 2.2.11: “Let $f: {\bf R}^2 \to {\bf R}$ be the function defined by $f(x,y) := x^2/y$ when $y \neq 0$, and $f(x,y) := 0$ when $y = 0$. Show that $\lim_{t \to 0} f(tx, ty) = f(0,0)$ for every $(x,y) \in {\bf R}^2$, but that $f$ is not continuous at the origin. Thus being continuous on every line through the origin is not enough to guarantee continuity at the origin!”
27. p. 34In Proposition 2.3.2, replace “Furthermore, ” with “Furthermore, if $X$ is non-empty”,
28. p. 37In Theorem 2.4.5, replace “Let $X$ be a subset…” with “Let $X$ be a non-empty subset…”.
29. p. 38In Exercise 2.4.7, “replace “every path-connected set” by “every non-empty path-connected set”. In Exercise 2.4.6, add the hypothesis that $I$ is non-empty. Exercise 2.4.2 can benefit from Theorem 2.4.6 and so should be moved to after Exercise 2.4.4.
30. p. 43Exercise 2.5.8 is incorrect (the space $\omega_1+1$ is sequentially compact) and should be deleted.
31. p. 44In Exercise 2.5.14, add “Hausdorff” before “topological space”.
32. p. 46In Definition 3.1.1, the domain of $f$ should be $E$ rather than $X$. Similarly for Proposition 3.1.5, Exercise 3.1.3, and Exercise 3.1.5. In Remark 3.1.2, $\lim_{x \in x_0; x \in E} f(x)$ should be $\lim_{x \to x_0; x \in E} f(x)$.
33. p. 47In Proposition 3.1.5(c), all occurrences of $\subset$ should be $\subseteq$.
34. p. 48In Exercise 3.1.1, add the hypothesis “Assume that $x_0$ is an adherent point of $E \backslash \{x_0\}$ (or equivalently, that $x_0$ is not an isolated point of $E$)”. In Exercise 3.1.3, replace the last three sentences with “If $X$ is a topological space and $Y$ is a Hausdorff topological space (see Exercise 2.5.4), prove the equivalence of Proposition 3.1.5(c) and 3.1.5(d) in this setting, as well as an analogue of Remark 3.1.6. What happens to these statements of $Y$ is not Hausdorff?”.
35. p. 52In the last paragraph of the section, $f|_Y$ should be $f|_E$.
36. p. 56In item (c) of Section 3.4, a right parenthesis is missing after Definition 3.2.1. In Definition 3.4.2, add “uniform metric” next to “sup norm metric” and $L^\infty$ metric”, and restrict the definition of $d_\infty$ to the case when $X$ is non-empty, then add “When $X$ is empty, we instead define $d_\infty(f,g)=0$“; similarly for Definition 3.5.5. In Remark 3.4.1, “(b) is a special case of (a)” should be “(a) is a special case of (b)”. Finally, in Definition 3.4.2, use $[0,+\infty)$ in place of ${\bf R}^+$.
37. p. 60In Example 3.5.8, “ratio test” should be “root test”, and Theorem 7.5.1 should be “from Analysis I”. Also “$f^{(n)}$ converges uniformly” should be “$\sum_{n=1}^\infty f^{(n)}$ converges uniformly”.
38. p. 61In the second to last display, the factor $2$ in front of $2 \varepsilon(b-a)$ should be deleted.
39. p. 62In Example 3.6.3, Lemma 7.3.3 should be “from Analysis I”.
40. p. 64At the end of the first paragraph, the period should be inside the parentheses.
41. p. ?In Remark 3.8.7, add “However, we will remark that the Dirac delta function can be viewed as an identity for the convolution operation that we will introduce shortly, after generalizing the notion of convolution to such functions.”
42. p. 76In the first display of Example 4.1.5, $(-2^n)$ should be $(-2)^n$.
43. p. 77In Remark 4.1.9, it is more appropriate to add “uniformly” after “assures us that the power series will converge”.
44. p. 78At the end of the Exercise 4.1.1, a right parenthesis should be added.
45. p. 79In Definition 4.2.4, add “with the property that every element of $E$ is a limit point of $E$” at the end of the first sentence. At the end of the second sentence, add “, in particular $f': E \to {\bf R}$ is also a function on $E$.”
46. p. 81In Corollary 4.2.12, “ecah” should be “each”.
47. p. 82In Exercise 4.2.3, the period should be inside the parentheses. In the first paragraph, a right parenthesis should be added.
48. p. 83At the end of Exercise 4.2.8(e), the period should be inside the parentheses. Also in the hint, Fubini’s theorem should be Theorem 8.2.2 of Analysis I, and a remark needs to be made that one may also need to study an analogue of the $d_m$ in which the $c_n$ are replaced by $|c_n|$. At the beginning of the exercise, “anaytic in $a$” should be “analytic at $a$“.
49. p. 86In the last two displays, $\limsup_{n \to \infty}$ should be $\limsup_{y \to 1; y \in [0,1)}$.
50. p. 91Before Definition 4.5.5, “exp is increasing” should be “exp is strictly increasing”.
51. p. 92At the end of Exercise 4.5.1, a right parenthesis should be added.
52. p. 99before the final paragraph, add “Inspired by Proposition 4.5.4, we shall use $\exp(z)$ and $e^z$ interchangeably. It is also possible to define $a^z$ for complex $z$ and real $a>0$, but we will not need to do so in this text.”
53. p. 102In the second paragraph parenthetical, the period should be inside the parentheses.
54. p. 103In the second paragraph, a period should be added before “In particular, we have…”.
55. p. 105In the last paragraph of Exercise 4.7.9, the period should be inside the parentheses (two occurrences).
56. p. 112In Example 5.2.6, $\sin(x)$ should be $\sin(2\pi x)$.
57. p. 113In Exercise 5.2.3, “so that” should be “show that”. For more natural logical flow, the placing of Exercises 5.2.2 and 5.2.4 should be swapped.
58. p. 116In Theorem 5.4.1, “trignometric” should be “trigonometric”. In the paragraph after Remark 5.3.8, the period should be inside the parenthesis. The triangle inequality for complex-valued Riemann integrals used in the proof has not been established previously, and should be stated and referenced as an exercise in this section (in two parts: first, to establish a triangle inequality for continuous real-valued functions, and then use the phase rotation trick $|z| = \sup_\theta \mathrm{Re} e^{i\theta} z$ to handle the complex-valued case.).
59. p. 125In the last sentence of Exercise 5.5.3, the period should be inside the parenthesis. In Exercise 5.5.4, add “Here the derivative of a complex-valued function is defined in exactly the same fashion as for real-valued functions.”
60. p. 129In Example 6.1.8, “clockwise” should be “counter-clockwise”.
61. p. 133At the end of the proof of Lemma 6.1.13, $1 \leq j \leq m$ should be $1 \leq j \leq n$. Expand the sentence “The composition of two linear transformations is again a linear transformation (Exercise 6.1.2).” to “The composition $T \circ S$ of two linear transformations $T,S$ is again a linear transformation (Exercise 6.1.2). It is customary in linear algebra to abbreviate such compositions $T \circ S$ of linear transformations by droppinng the $\circ$ symbol, thus $T \circ S = TS$.”
62. p. 134In Lemma 6.2.1, “$x_0 \in E$, and $L \in {\bf R}$” should be “$L \in {\bf R}$, and let $x_0$ be a limit point of $E$“. In the previous display, $E \backslash \{x_0\}$ should be $E - \{x_0\}$.
63. p. 135In the first paragraph, the period should be inside the parenthesis. In Definition 6.2.2, $x_0$ should be a limit point of $E$.
64. p. 138In Example 6.3.3, “the left derivative” should be “the negative of the left derivative”. In the last sentence, the period should be inside the parenthesis.
65. p. 139In the second paragraph, second sentence, the period should be inside the parenthesis; also in the final sentence. Expand the third display to “$\frac{\partial f}{\partial x_j}(x_0) = D_{e_j} f(x_0) = -D_{-e_j} f(x_0) = f'(x_0) e_j$, and expand “From Lemma 6.3.5” to “From Lemma 6.3.5 (and Proposition 9.5.3 from Analysis I)”.
66. p. 140In the beginning of the proof of Theorem 6.3.8, $L(v_j)_{1 \leq j \leq m}$ should be $L(v_j)_{1 \leq j \leq n}$, and similarly the sum on the RHS should be from $1$ to $n$ rather than from $1$ to $m$. “Because each partial derivative … is continuous on $F$” should be “Because each partial derivative … exists on $F$ and is continuous at $x_0$“.
67. p. 141The period in the last line (before “and so forth”) should be deleted.
68. p. 142At the end of the page, $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{i=1}^m$ should be $(\sum_{j=1}^n v_{j}\frac{\partial f_i}{\partial x_j}(x_0))_{1\leq i\leq m}$.
69. p. 144In Exercise 6.3.2, $D_{e_j} f(x_0) = D_{-e_j} f(x_0)$ should be $D_{e_j} f(x_0) = -D_{-e_j} f(x_0)$.
70. p. 146In the second paragraph, third sentence, the period should be inside the parenthesis.
71. p. 148In the proof of Clairaut’s theorem, $|x| \leq 2\delta$ should be $\| x\| \leq 2\delta$.
72. p. 151In Exercise 6.6.1, the range of $f$ should be $[a,b]$ rather than ${\bf R}$.
73. p. 153In the second paragraph of the proof of Theorem 6.7.2, “$f(x_0)$ is not invertible” should be “$f'(x_0)$ is not invertible”.
74. p. 154In the last text line, $f(x)-x$ can be $g(x)=f(x)-x$ for clarity.
75. p. 155In the proof of Theorem 6.7.2, after the display after “we have by the fundamental theorem of calculus. add “where the integral of a vector-valued function is defined by integrating each component separately.”
76. p. 156$V - 0$ should be $V - \{0\}$. The definition of $U$ should be $f^{-1}(V) \cap B(0,r)$ rather than $f^{-1}(B(0,r/2))$ (and the later reference to $U = f^{-1}(V)$ can be replaced just by $U$).
77. p. 157In the final paragraph of Section 6.7, “differentiable at $x_0$” should be “differentiable at $f(x_0)$“. Add the following Exercise 6.7.4 after Exercise 6.7.3: “Let the notation and hypotheses be as in Theorem 6.7.2. Show that, after shrinking the open sets $U, V$ if necessary (while still having $x_0 \in U$, $f(x_0) \in V$ of course), the derivative map $f'(x)$ is invertible for all $x \in U$, and that the inverse map $f^{-1}$ is differentiable at every point of $V$ with $(f^{-1})'(f(x)) = (f'(x))^{-1}$ for all $x \in U$. Finally, show that $f^{-1}$ is continuously differentiable on $V$.”
78. p. 158In the first paragraph, final sentence, the period should be inside the parentheses.
79. p. 161Add the following Exercise 6.8.1: “Let the notation and hypotheses be as in Theorem 6.8.1. Show that, after shrinking the open sets $U,V$ if necessary , that the function $g$ becomes continuously differentiable on all of $U$, and the equation (6.1) holds at all points of $U$.”
80. p. 163after “if $A$ and $B$ are disjoint”, add “, and more generally, that $m(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^\infty m(A_n)$ when $A_1,A_2,\dots$ are disjoint”.
81. p. 164In the first paragraph of Section 7.1, $\Omega \subset {\bf R}^n$ should be $\Omega \subseteq {\bf R}^n$.
82. p. 165Superfluous period in Theorem 7.1.1. “Since everything is positive” should be “Since everything is non-negative”, and in the preceding sentence, add “; for instance, in this chapter we adopt the convention that an infinite sum $\sum_{j \in J} a_j$ of non-negative quantities $a_j$ is equal to $+\infty$ if the sum is not absolutely convergent.”
83. p. 169After (7.1), $\prod_{j=1}^n [a_i,b_i]$ should be $\prod_{i=1}^n [a_i,b_i]$.
84. p. 170In the first paragraph “For all other values if $x$” should be “For all other values of $x$“.
85. p. 172$\subset$ should be $\subseteq$ (three occurrences). In Example 7.2.9, “each rational number ${\bf Q}$” should be “each rational number $q$“.
86. p. 173In Exercise 7.2.2, final sentence: period should be inside parentheses. Also, add “Here we adopt the convention that $c \times +\infty = +\infty \times c$ is infinite for any $0 < c \leq +\infty$ and vanishes for $c = 0$.” In Example 7.2.12, “countable additivity” should be “countable sub-additivity”.
87. p. 174In the penultimate paragraph, “identical or distinct” should be “identical or disjoint”, and $\subset$ should be $\subseteq$. Also, “coset of ${\bf R}$” should be “coset of ${\bf Q}$“; in the next paragraph, “the rationals ${\bf R}$” should be “the rationals ${\bf Q}$“. In Exercise 7.2.5, “Q1” should be “Exercise 7.2.3”.
88. p. 175In the second paragraph, “constrution” should be “construction”. After the third paragraph, add “Note also that the translates $q+E$ for $q \in {\bf Q}$ are all disjoint. For, if there were two distinct $q,q' \in {\bf Q}$ with $q+E$ intersecting $q'+E$, then there would be $A,A' \in {\bf R}/{\bf Q}$ such that $q+x_A = q'+x_{A'}$. But then $A = x_A + {\bf Q} = x_{A'} + {\bf Q} = A'$ and thus $x_A = x_{A'}$, which implies $q=q'$, contradicting the hypothesis.”
89. p. 176In the proof of Proposition 7.3.3, “cardinality 3n” should be “cardinality $3n$“.
90. p. 178In Lemma 7.4.5, “and any set $A$” should be “then for any set $A$“.
91. p. 180In the first paragraph, “Lemma 7.4.5” should be “Lemma 7.4.4(d)”. Also, in the display preceding this paragraph, enclose the sum in parentheses in the middle and right-hand sides (so that the supremum is taken over the sum rather than just the first term).
92. p. 181“… on our wish list is (a)” should be “… on our wish list is (i)”.
93. p. 187In Example 8.1.2, the period should be inside the parentheses in the first parenthetical, and the final right parenthsis should be deleted.
94. p. 188In Lemma 8.1.5 the function $f$ should take values in ${}[0,+\infty]$ rather than ${\bf R}$ (and then the requirement that $f$ be non-negative can be deleted).
95. p. 189In the parenthetical sentence before Remark 8.1.8, the period should be inside the parentheses. In the first display in Lemma 8.1.9, the right-hand side summation should be up to $N$ rather than $n$, and “are a finite number” should be “be a finite number”. In Example 8.1.7, “the integral” should be “the interval”.
96. p. 190In the final display in the proof of Lemma 8.1.9, an equals sign should be inserted to the left of the final line.
97. p. 194In Theorem 8.2.9, $f_n$ should take values in $[0,+\infty]$ rather than ${\bf R}$.
98. p. 196Before the second display, Proposition 8.2.6(cdf) should be Proposition 8.2.6(bce). Also add “It is not difficult to check that the $E_n$ are measurable”. In the first paragraph, all instances of $\subset$ should be $\subseteq$.
99. p. 197After the third display. Proposition 8.1.9(b) should be Proposition 8.1.10(bd).
100. p. 199Exercise 8.2.4 should be moved to Section 8.3 (as it uses the absolutely convergent integral).
101. p. 200In the hint to Exercise 8.2.10, the “for all $n \geq N$” should be moved inside the set builder notation $\{ x \in [0,1]: f_n(x) > 1/m \}$, thus using $\{ x \in [0,1]: f_n(x) > 1/m \hbox{ for all } n \geq N \}$ instead.
102. p. 201Before Definition 8.3.2, when Corollary 7.5.6 is invoked, add “(which can be extended to functions taking values in ${\mathbf R}^*$ without difficulty)”.
103. p. 202In the start of the proof of Theorem 8.3.4, add “If $F$ was infinite on a set of positive measure then $F$ would not be absolutely integrable; thus the set where $F$ is infinite has zero measure. We may delete this set from $\Omega$ (this does not affect any of the integrals) and thus assume without loss of generality that $F(x)$ is finite for every $x\in \Omega$, which implies the same assertion for the $f_n(x)$.
104. p. 204In the second display, $\leq A - \frac{1}{n}$ should be $\geq A + \frac{1}{n}$ instead.
105. p. 205In Proposition 8.4.1, add the hypothesis that $I$ is bounded.
106. p. 206In the last paragraph, last sentence, the period should be inside the parentheses. In the last two displays, $\Omega$ should be $I$.
107. p. 207In the third paragraph, “Secondly, we could fix” should be “Thirdly, we could fix”.
108. p. 208In the last paragraph, Lemma 8.1.4 should be Lemma 8.1.5.

Second edition (hardcover)

27 corrections

1. p. 351At the end of Example 12.1.6, add “Extending the convention from Example 12.1.4, if we refer to ${\bf R}^n$ as a metric space, we assume that the metric is given by the Euclidean metric unless otherwise specified.”
2. p. 372In Case 1 of the proof of Theorem 12.5.8, all occurrences of “$y^{(n)}$ should be $y^{(n_j)}$ in the second paragraph.
3. p. 374In Exercise 12.5.12(b), the phrase “with the Euclidean metric” should be deleted.
4. p. 390In Exercise 13.5.5, “there exist $a, b \in X$ such that the “interval” $\{ y \in X: a < y < b \}$” should be replaced with “there exists a set $I$ which is an interval $\{y \in X: a < y < b\}$ for some $a, b \in X$, a ray $\{y \in X: a < y \}$ for some $a \in X$, the ray $\{ y \in X: y < b\}$ for some $b \in X$, or the whole space $X$, which”. In Exercises 13.5.6 and 13.5.7, “Hausdorff” should be “not Hausdorff”.
5. p. 390Exercise 13.5.8 should be replaced as follows: “Show that there exists an uncountable well-ordered set $\omega_1+1$ that has a maximal element $\infty$, and such that the initial segments $\{ x \in \omega_1+1: x < y \}$ are countable for all $y \in \omega_1+1 \backslash \{\infty\}$. (Hint: Well-order the real numbers using Exercise 8.5.19, take the union of all the countable initial segments, and then adjoin a maximal element $\infty$.) If we give $\omega_1+1$ the order topology (Exercise 13.5.5), show that $\omega_1+1$ is compact; however, show that not every sequence has a convergent subsequence.”
6. p. 395In Proposition 14.1.5(d), add “Furthermore, if $x_0 \in E$, then $f(x_0)=L$.”
7. p. 396In Exercise 14.1.5, $\lim_{y \to y_0; y \in f(E)} g(x)$ should be $\lim_{y \to y_0; y \in f(E)} g(y)$, and $\lim_{x \to x_0; x \in E} g \circ f(x_0)$ should be $\lim_{x \to x_0; x \in E} g \circ f(x)$.
8. p. 425In Theorem 15.1.6(d), the summation should start from n=1 rather than n=0.
9. p. 427Just before Definition 15.2.4, “for some $a \in {\ bf R}$” should be “for some $r > 0$“.
10. p. 431In Exercise 15.2.8(e), “$d_m (x-b)^n$” should be “$d_m (x-b)^m$. In Exercise 15.2.8(d), $(s-\varepsilon)^m$ should be $(s-\varepsilon)^{-m}$.
11. p. 433(proof of Theorem 15.3.1): $s_N$ in the third display and $s_0$ in the next line should be$S_N$ and $S_0$ respectively.
12. p. 452In Exercise 15.7.2, $x-y$ should be $x_0-y$. In Exercise 15.7.6, “complex real number” should be “complex number”.
13. p. 473In Exercise 16.5.4, $in \hat f(n)$ should be $2\pi i n \hat f(n)$.
14. p. 477In Example 17.1.7, $x \in {\Bbb R}$ should be $c \in {\Bbb R}$.
15. p. 486In Definition 17.3.7, $1 \leq j \leq m$ should be $1 \leq j \leq n$, and $x_0+tv$ should be $x_0+te_j$.
16. p. 488In the definition of L in the proof of Theorem 17.3.8, m should be n.
17. p. 492In Exercise 17.3.1, Exercise 17.1.3 should be Exercise 17.2.1.
18. p. 495In the proof of Theorem 17.5.4, $a'$ should equal $\frac{\partial}{\partial x_i} \frac{\partial}{\partial x_j} f(0)$ rather than $\frac{\partial}{\partial x_j} \frac{\partial}{\partial x_i} f(0)$.
19. p. 499proof of Lemma 17.6.6: After “$F$ does indeed map $B(0,r)$ to itself.”, add “The same argument shows that for a sufficiently small $\varepsilon > 0$, $F$ maps the closed ball $\overline{B(0,r-\varepsilon)}$ to itself. After “$F$ is a strict contraction”, add “on $B(0,r)$, and hence on the complete space $\overline{B(0,r-\varepsilon)}$“.
20. p. 502proof of Theorem 17.7.2: “$f^{-1}(y)={\tilde f}^{-1}(y+f(x_0))$” should be “$f^{-1}(y)={\tilde f}^{-1}(y-f(x_0))$“.
21. p. 505Section 17.8: $(0,1)$ should be $(-1,0)$. In the second paragraph, the function $f$ should be $g$ (for better compatibility with the discussion of the implicit function theorem).
22. p. 508proof of Theorem 17.8.1, “U is open and contains $(y_1, \dots,y_{n-1},0)$” should be “U is open and contains $(y_1, \dots,y_{n-1})$“.
23. p. 515In the display before Definition 18.2.4, $j=\in J$ should be $j \in J$. In Definition 18.2.4, $\sum_{j=1}^\infty$ should be $\sum_{j \in J}$.
24. p. 520In Example 18.2.9, $\sum_{q\in\mathbf{Q}} m*(\mathbf{Q})$ should be $\sum_{q\in\mathbf{Q}} m*(\{q\})$ in the display.
25. p. 528proof of Lemma 18.4.8: On the second line, “let $A$ be any other measurable set” should be “let $A$ be an arbitrary set (not necessarily measurable)”.
26. p. 545In Corollary 19.2.11, “non-negative functions” should be “non-negative measurable functions”.
27. p. 555Remark 19.5.2: x and y should be swapped in “equals 1 when $x>0$ and y=0, equals -1 when $x<0$ and y=0, and equals zero otherwise”.

First edition (softcover)

42 corrections

1. p. 392example 12.1.7: $5+2=7$ should be $3+4=7$.
2. p. 393example 12.1.9: $\sup(5,2)=7$ should be $\sup(3,4)=4$.
3. p. 394example 12.1.13: (iii) should be (c).
4. p. 403example 12.2.13: delete the redundant “, but not the other”.
5. p. 404line 4: “neither open and closed” should be “neither open nor closed”.
6. p. 415line 3: $\alpha \in A$ should be $\alpha \in I$.
7. p. 416line 11: “$k \geq j$” should be “$k > j$“.
8. p. 419line -2: In Exercise 12.5.15, = should be $\neq$. Also, “that by counterexample” should be “by counterexample that”
9. p. 426Exercise 13.2.9: $X$ should be ${\Bbb R}$ throughout. Also, the definition of limsup and liminf for functions has not been given; it can be reviewed here, e.g. by inserting “where we define $\limsup_{x \to x_0} f(x) := \inf_{r>0} \sup_{x: |x-x_0| \leq r} f(x)$ and $\liminf_{x \to x_0} f(x) := \sup_{r>0} \inf_{x: |x-x_0| \leq r} f(x)$.”
10. p. 435Definition 13.5.6: “metric space” should be “topological space”.
11. p. 438Exercise 13.5.9: One needs to assume as an additional hypothesis that X is first countable, which means that for every x in X there exists a countable sequence V_n of neighborhoods of x, such that every neighbourhood of x contains one of the V_n.
12. p. 452Exercise 14.3.6: “Propositoin” should be “Proposition”.
13. p. 452Exercise 14.3.8: “$x \in {\Bbb R}$” should be “$x \in X$“.
14. p. 458Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
15. p. 459In line 11, $2 \epsilon$ should be $\epsilon$.
16. p. 464) missing at the end of Exercise 14.7.2. An additional exercise, Exercise 14.7.3 is missing; it should state “Prove Corollary 14.7.3.”.
17. p. 466Exercise 14.8.8 should be Exercise 14.8.2.
18. p. 467Exercise 14.8.11 should be Exercise 14.8.4.
19. p. 469“Limits of integration” should be “Limits of summation”. In Lemma 14.8.14, $3M+2\delta$ should be $1+4M$, and Exercise 14.8.14 should be Exercise 14.8.6.
20. p. 470Exercise 14.8.15 should be Exercise 14.8.7. Exercise 14.8.16 should refer to a (currently non-existent) Exercise 14.8.9, which of course would be to prove Lemma 14.8.16.
21. p. 471At the end of the proof of Corollary 14.8.19, $|P(y)-g(y)| \leq \varepsilon$ should be $|P(y)-f(y)| \leq \varepsilon$.
22. p. 472In Exercise 14.8.2(c), Lemma 14.8.2 should be Lemma 14.8.8.
23. p. 477In Exercise 15.1.1(e), Corollary 14.8.18 should be Corollary 14.6.2.
24. p. 478In Example 15.2.2, $\frac{1}{x-1}$ should be $\frac{1}{1-x}$.
25. p. 482In Exercise 15.2.5, the $(x-b)^n$ on the right-hand side should be $(x-b)^m$.
26. p. 486In second and third display, y should be in ${}[0,1]$ rather than $(-1,1)$.
27. p. 493In Exercise 15.5.4, $x < 0$ should be $x \leq 0$.
28. p. 501In Theorem 17.7.2, “if $f(x_0)$ is not invertible” should be “if $f'(x_0)$ is not invertible”.
29. p. 502In Exercise 15.6.6, Lemma 15.6.6 should be Lemma 15.6.11.
30. p. 511“Fourier… was, among other things, the governor of Egypt during the reign of Napoleon. After the Napoleonic wars, he returned to mathematics.” should be “Fourier… was, among other things, an administrator accompanying Napoleon on his invasion of Egypt, and then a Prefect in France during Napoleon’s reign.”
31. p. 556In Theorem 17.5.4, f can take values in ${\Bbb R}^m$ and not just in $R$; insert the line “By working with one component of $f$ at a time, we may assume $m=1$” as the first line of the proof. Also, $f(x-x_0)$ should be $f(x+x_0)$.
32. p. 557In the second display, $-f(0)$ should be $+f(0)$.
33. p. 560In Exercise 17.6.1, add the hypothesis “and $f'$ is continuous” before “, show that $f$ is a strict contraction”.
34. p. 561In Exercise 17.6.3, change “which is a strict contraction” to “such that $|f(x)-f(y)| < |x-y|$ for all distinct $x,y$ in ${}[a,b]$“. In Exercise 17.6.8, $\max(c,c')$ should be $\min(c,c')$.
35. p. 562In Theorem 17.7.2, $T: E \to {\Bbb R}^n$ should be $f: E \to {\Bbb R}^n$.
36. p. 565line -7: $U$ should be $f^{-1}(B(0,r/2))$ rather than $f^{-1}(B(0,r))$.
37. p. 570first display: all partial derivatives should have a – sign (not just the first one). Last paragraph: “Thus $(x_1,\ldots,x_{n-1})$ lies in W” should be “Thus $(x_1,\ldots,x_{n-1})$ lies in U”.
38. p. 571second display: add “$=0$” at the end.
39. p. 584Corollary 18.2.7: “$x_i \in [a_i,b_i]$” should be “$x_i \in (a_i,b_i)$“.
40. p. 599Definition 18.5.9: $(a,\infty)$ should be $(a,+\infty]$.
41. p. 600In Lemma 18.5.10, $f: \Omega \to {\Bbb R}$ should be $f: \Omega \to {\Bbb R}^*$. In the second and fourth lines of the proof of this lemma, $(a,+\infty)$ should be $(a,+\infty]$.
42. p. 616-617Exercise 19.2.10: ${\Bbb R}$ should be ${}[0,1]$ throughout.