Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 274 pages. ISBN 81-85931-62-3

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 —

  • 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. 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 —

  • p. 546: 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. 548: In the definition of L in the proof of Theorem 17.3.8, m should be n.
  • p. 625, 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”.

Thanks to Carlos, EO, Florian, Eric Koelink, Wang Kunyang, Jason M., Manoranjan Majji, Geoff Mess, Cristina Pereyra, Haokun Xu, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.