Analysis, Volume I
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 422 pages. ISBN 81-85931-62-3

This is basically an expanded and cleaned up version of my lecture notes for Math 131A. 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 II.  A second edition is being prepared.

There are no solution guides for this text.

  • Sample chapters (contents, natural numbers, set theory, integers and rationals, logic, decimal system, index)

— Errata —

  • p.5, line 6 from bottom: \sin(\pi/2-2) should be \sin(\pi/2-z).
  • p. 59, Lemma 3.3.12: f should map Z to W, and h should map X to Y. In the proof of this lemma: g \circ h is a function from X to Z, and f \circ g is a function from Y to W.
  • p. 67, last paragraph: \alpha \in A should be \alpha \in I.
  • p. 98: In Exercise 4.2.1, Corollary 2.3.7 should be Corollary 4.1.9.  In Exercise 4.2.6, x,y,z should be rational numbers, not real.
  • p. 101: In Definition 4.3.9, after “x^0 := 1“, add “; in particular, we define 0^0 := 1“.
  • p. 127: In Exercise 5.3.4: add “(Hint: use Exercise 5.2.2.)”.
  • p. 131, line 12 from bottom: “they cannot be than” should be “they cannot be larger than”.
  • p. 175, Exercise 6.6.3: In the hint, replace “introduce” by “recursively introduce”, and insert “; n > n_{j-1}” after “|a_n| \geq j” (two occurrences), with the parenthetical “(omitting the n > n_{j-1} condition when j=0)” inserted after the recursive definition of n_j.
  • p. 197, in second line of proof of Proposition 7.3.4: the second sum should be \sum_{k=0}^\infty rather than \sum_{k=0}^K.
  • p. 216, Exercise 8.1.9: It needs to be noted that this exercise requires the axiom of choice from Section 8.4.
  • p. 220, Lemma 8.2.5: It needs to be noted that this lemma requires the axiom of choice from Section 8.4. Similarly, the case in Proposition 8.2.6 in which X is uncountable requires the axiom of choice also.
  • p. 227, Exercise 8.3.2: g(x) := f(x) should be g(x) := f^{-1}(x).
  • p. 236, last line: “for any good set Y’” should be “for any good set Y’ with A \cap Y' non-empty”.
  • p. 255, Proposition 9.3.9(b): f(x_0) should be L.
  • p. 303, Exercise 10.4.3(a): The limit should be in the set (0,\infty) \backslash \{1\} rather than (0,\infty).
  • p. 336, line 13: replace “we have made no assumption on \alpha” with “the function \alpha: {\Bbb R} \to {\Bbb R} could have been arbitrary”.
  • p. 337, Exercise 11.8.1: Lemma 11.8.1 should be Lemma 11.8.4.
  • p. 337, Exercise 11.8.5: In the last display, f(0) should be 2f(0).
  • p. 342, Exercise 11.9.1: “the function f is not differentiable” should be “the function F(x) := \int_{[0,x]} f is not differentiable.
  • p. 383, first display: a_n \times \hbox{ten}^i should be a_n \times \hbox{ten}^n.
  • p. 387, fourth display: a_n should be a_{n+1}.

— Errata for the second edition —

  • p. xii, bottom: “solidifed” -> “solidified”.
  • p. xiv, top: “to know how to to” –> “to know how to”.
  • p. 51, Exercise 3.1.1: (3.1.4) should be Definition 3.1.4.
  • p. 79, 5th line of proof of Lemma 3.6.9: 1 \leq i \leq N should be 1 \leq i \leq n.
  • p.80, 6th line of proof of Proposition 3.6.8: Proposition 3.6.4 should be Lemma 3.6.9.
  • p. 157, Theorem 5.2.11(b), (c): Replace “Suppose that M” with “Suppose that M \in {\Bbb R}^*” (two occurrences).
  • p. 157, Exercise 6.2.2: Proposition 6.2.11 should be Theorem 6.2.11.
  • p. 175, Exercise 6.6.5: Replace “the formula n_j := \min\{n \in {\Bbb N}: |a_n-L| \leq 1/j\}, explaining why the set \{n \in {\Bbb N}: |a_n-L| \leq 1/j\} is non-empty” with “the recursive formula n_j := \min\{n > n_{j-1}: |a_n-L| \leq 1/j\}, with the convention n_0=0, explaining why the set \{n > n_{j-1}: |a_n-L| \leq 1/j\} is non-empty”.
  • p. 195, Exercise 7.2.6: Add “How does the proposition change if we assume that a_n does not converge to zero, but instead converges to some other real number L?”
  • p. 215, Exercise 8.1.1: This exercise requires the axiom of choice, Axiom 8.1.
  • p. 227, Exercise 8.3.2: f should be an injection rather than a bijection.  In the definition of g, \bigcup_{n=0}^\infty D_n should be \bigcup_{n=1}^\infty D_n (two occurrences).
  • p. 231, Exercise 8.4: y \in y should be y \in Y.
  • p. 247, Lemma 9.1.21.  One needs the additional hypothesis “We assume that a<b.”
  • p. 291, Proposition 10.1.7: One needs the additional hypothesis x_0 \in X.  Similarly for Proposition 10.1.10, theorem 10.1.13, and Proposition 10.3.1.
  • p. 292, Definition 10.1.11: “For every x_0 \in X” should be “For every limit point x_0 \in X\".
  • p. 293, Remark 10.1.14: Leibnitz should be Leibniz.
  • p. 296, Definition 10.2.1: x \in X should be x_0 \in X.

Thanks to Evangelos Georgiadis, Erik Koelink, Manoranjan Majji, Pieter Naaijkens, Cristina Pereyra, Tim Reijnders, Pieter Roffelsen, Luke Rogers, Marc Schoolderman, Daan Wanrooy, Yandong Xiao, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.