Last updated May 9, 2019

Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006.  Third edition, 2014
Hardcover, 236 pages.

ISBN 81-85931-62-3 (first edition)

This is basically an expanded and cleaned up version of my lecture notes for Math 131B. In the US, it is available through the American Mathematical Society. It is part of a two-volume series; here is my page for Volume I.

— Errata to the first edition (softcover) —

  • p. 392, example 12.1.7: 5+2=7 should be 3+4=7.
  • p. 393, example 12.1.9: \sup(5,2)=7 should be \sup(3,4)=4.
  • p. 394, example 12.1.13: (iii) should be (c).
  • p. 403, example 12.2.13: delete the redundant “, but not the other”.
  • p. 404, line 4: “neither open and closed” should be “neither open nor closed”.
  • p. 415, line 3: \alpha \in A should be \alpha \in I.
  • p. 416, line 11: “k \geq j” should be “k > j“.
  • p. 419, line -2: In Exercise 12.5.15, = should be \neq. Also, “that by counterexample” should be “by counterexample that”
  • p. 426, Exercise 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).”
  • p. 435, Definition 13.5.6: “metric space” should be “topological space”.
  • p. 438, Exercise 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.
  • p. 452, Exercise 14.3.6: “Propositoin” should be “Proposition”.
  • p. 452, Exercise 14.3.8: “x \in {\Bbb R}” should be “x \in X“.
  • p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
  • p. 459: In line 11, 2 \epsilon should be \epsilon.
  • 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.”.
  • p. 466: Exercise 14.8.8 should be Exercise 14.8.2.
  • p. 467: Exercise 14.8.11 should be Exercise 14.8.4.
  • 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.
  • p. 470: Exercise 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.
  • p. 471: At the end of the proof of Corollary 14.8.19, |P(y)-g(y)| \leq \varepsilon should be |P(y)-f(y)| \leq \varepsilon.
  • p. 472: In Exercise 14.8.2(c), Lemma 14.8.2 should be Lemma 14.8.8.
  • p. 477: In Exercise 15.1.1(e), Corollary 14.8.18 should be Corollary 14.6.2.
  • p. 478: In Example 15.2.2, \frac{1}{x-1} should be \frac{1}{1-x}.
  • p. 482: In Exercise 15.2.5, the (x-b)^n on the right-hand side should be (x-b)^m.
  • p. 486: In second and third display, y should be in {}[0,1] rather than (-1,1).
  • p. 493: In Exercise 15.5.4, x < 0 should be x \leq 0.
  • p. 501: In Theorem 17.7.2, “if f(x_0) is not invertible” should be “if f'(x_0) is not invertible”.
  • p. 502: In Exercise 15.6.6, Lemma 15.6.6 should be Lemma 15.6.11.
  • 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.”
  • p. 556: In 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).
  • p. 557: In the second display, -f(0) should be +f(0).
  • p. 560: In Exercise 17.6.1, add the hypothesis “and f' is continuous” before “, show that f is a strict contraction”.
  • p. 561: In 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').
  • p. 562: In Theorem 17.7.2, T: E \to {\Bbb R}^n should be f: E \to {\Bbb R}^n.
  • p. 565, line -7: U should be f^{-1}(B(0,r/2)) rather than f^{-1}(B(0,r)).
  • p. 570, first 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”.
  • p. 571, second display: add “=0” at the end.
  • p. 584, Corollary 18.2.7: “x_i \in [a_i,b_i]” should be “x_i \in (a_i,b_i)“.
  • p. 599, Definition 18.5.9: (a,\infty) should be (a,+\infty].
  • p. 600: In 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].
  • p. 616-617, Exercise 19.2.10: {\Bbb R} should be {}[0,1] throughout.

— Errata to the second edition (hardcover) —

  • p. 351, At 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.”
  • p. 372, In Case 1 of the proof of Theorem 12.5.8, all occurrences of “y^{(n)} should be y^{(n_j)} in the second paragraph.
  • p. 374, In Exercise 12.5.12(b), the phrase “with the Euclidean metric” should be deleted.
  • p. 390: In 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”.
  • p. 390: Exercise 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.”
  • p. 395: In Proposition 14.1.5(d), add “Furthermore, if x_0 \in E, then f(x_0)=L.”
  • p. 396: In 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).
  • p. 425: In Theorem 15.1.6(d), the summation should start from n=1 rather than n=0.
  • p. 427: Just before Definition 15.2.4, “for some a \in {\ bf R}” should be “for some r > 0“.
  • p. 431: In 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}.
  • 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.
  • p. 452: In Exercise 15.7.2, x-y should be x_0-y. In Exercise 15.7.6, “complex real number” should be “complex number”.
  • p. 473: In Exercise 16.5.4, in \hat f(n) should be 2\pi i n \hat f(n).
  • p. 477: In Example 17.1.7, x \in {\Bbb R} should be c \in {\Bbb R}.
  • p. 486: In 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.
  • p. 488: In the definition of L in the proof of Theorem 17.3.8, m should be n.
  • p. 492: In Exercise 17.3.1, Exercise 17.1.3 should be Exercise 17.2.1.
  • p. 495: In 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).
  • p. 499, proof 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)}“.
  • p.502, proof 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))“.
  • p. 505, Section 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).
  • p.508, proof 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})“.
  • p. 515: In 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}.
  • p. 520: In Example 18.2.9, \sum_{q\in\mathbf{Q}} m*(\mathbf{Q}) should be \sum_{q\in\mathbf{Q}} m*(\{q\}) in the display.
  • p. 528, proof 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)”.
  • p. 545: In Corollary 19.2.11, “non-negative functions” should be “non-negative measurable functions”.
  • p. 555, Remark 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”.

— Errata for the third edition (hardcover) —

  • Page 18: The proof of Lemma 1.4.3 should refer to Exercise 1.4.1, not Exercise 1.4.3.
  • Page 22: In Definition 1.5.3, add “We call (X,d) bounded if X is bounded”.
  • Page 32: In 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”.
  • Page 42: In Exercise 2.5.5, V \subset X should be V \subseteq X. In Exercise 2.5.13, “topological space” should be “Hausdorff topological space”.
  • Page 48: In Exercise 3.1.5, \lim_{x \in x_0; x \in E} should be \lim_{x \to x_0; x \in E}.
  • Page 71: In Corollaries 3.8.18 and 3.8.19, “supported on [0,1]” is redundant and may be deleted.
  • Page 98, 100: 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.”
  • Page 122: In Remark 5.5.2, “continuously differentiable” may be relaxed to just “differentiable”, and “twice continuously differentiable” may be relaxed to “continuously differentiable”.
  • Page 123: In the sixth line, “Corollary 5” should be “Corollary 5.3.6”.
  • Page 140: In the last paragraph, “continuous on F” should be “continuous at x_0“.
  • Page 146: The definition of continuous differentiability needs to be detached from Definition 6.5.1 and placed near the beginning of this page.
  • Page 159: In the first line of the third display in the proof of Theorem 6.8.1, the f‘s should be F‘s.
  • Page 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”.
  • Page 176: In Proposition 7.3.3, \sum_{q \in J} q+E should be \bigcup_{q \in J} q+E (two occurrences).
  • Page 178: In Corollary 7.4.7, m(B \backslash A) = m(B) - m(A) should be m(B \backslash A) + m(A) = m(B).

— Errata for the corrected third edition —

  • Page 10: In Exercise 1.1.8, a right parenthesis is missing at the end of the last sentence.
  • Page 16: In 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.
  • Page 22: In Definition 1.5.3, insert “for every x \in X” before “there exists a ball” (in order to keep the empty metric space bounded).
  • Page 23: In 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.
  • Page 33: Add 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!”
  • Page 43: Exercise 2.5.8 is incorrect (the space \omega_1+1 is sequentially compact) and should be deleted.
  • Page 46: In 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.
  • Page 48: In 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)”.
  • Page 56: In item (c) of Section 3.4, a right parenthesis is missing after Definition 3.2.1.
  • Page 64: At the end of the first paragraph, the period should be inside the parentheses.
  • Page 78: At the end of the Exercise 4.1.1, a right parenthesis should be added.
  • Page 82: In Exercise 4.2.3, the period should be inside the parentheses.  In the first paragraph, a right parenthesis should be added.
  • Page 83: At the end of Exercise 4.2.8(e), the period should be inside the parentheses.
  • Page 92: At the end of Exercise 4.5.1, a right parenthesis should be added.
  • Page 102: In the second paragraph parenthetical, the period should be inside the parentheses.
  • Page 103: In the second paragraph, a period should be added before “In particular, we have…”.
  • Page 105: In the last paragraph of Exercise 4.7.9, the period should be inside the parentheses (two occurrences).
  • Page 113: In Exercise 5.2.3, “so that” should be “show that”.
  • Page 116: In Theorem 5.4.1, “trignometric” should be “trigonometric”.  In the paragraph after Remark 5.3.8, the period should be inside the parenthesis.
  • Page 125: In 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.”
  • Page 129: In Example 6.1.8, “clockwise” should be “counter-clockwise”.
  • Page 133: At the end of the proof of Lemma 6.1.13, 1 \leq j \leq m should be 1 \leq j \leq n.
  • Page 135: In the first paragraph, the period should be inside the parenthesis.
  • Page 138: In 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.
  • Page 139: In the second paragraph, second sentence, the period should be inside the parenthesis; also in the final sentence.
  • Page 140: In 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 m rather than from 1 to n.
  • Page 141: The period in the last line (before “and so forth”) should be deleted.
  • Page 142: At 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}.
  • Page 144: In 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).
  • Page 146: In the second paragraph, third sentence, the period should be inside the parenthesis.
  • Page 148: In the proof of Clairaut’s theorem, |x| \leq 2\delta should be \| x\| \leq 2\delta.
  • Page 152: In Exercise 6.6.8, \min(c,c') should be \max(c,c').
  • Page 154: In the last text line, f(x)-x can be g(x)=f(x)-x for clarity.
  • Page 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).
  • Page 157: 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.”
  • Page 158: In the first paragraph, final sentence, the period should be inside the parentheses.
  • Page 161: Add 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.”
  • Page 173: In 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.”
  • Page 187: In Example 8.1.2, the period should be inside the parentheses in the first parenthetical, and the final right parenthsis should be deleted.
  • Page 189: In the parenthetical sentence before Remark 8.1.8, the period should be inside the parentheses.
  • Page 206: In the last paragraph, last sentence, the period should be inside the parentheses.

Caution: the page numbering is not consistent across editions.  In the third edition, the chapters were renumbered to start from 1, rather than from 12.

Thanks to Quentin Batista, Biswaranjan Behera, José Antonio Lara Benítez, Petrus Bianchi, Carlos, EO, Florian, Aditya Ghosh, Gökhan Güçlü, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Matthis Lehmkühler, Rami Luisto, Jason M., Manoranjan Majji, Geoff Mess, Cristina Pereyra, SkysubO, Kent Van Vels, Xueping, Sam Xu, Zelin, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.