Last updated: Aug 26, 2016

Analysis, Volume I
Terence Tao
Hindustan Book Agency, January 2006.  Third edition, 2014
Hardcover, 368 pages.

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

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.  It is currently in its third edition.

There are no solution guides for this text.

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

— Errata —

  • p. 2, item 3: “can you add” should be “Can you add”.
  • p. 9, line 5: “right-hand side” should be “left-hand side”.
  • p. 10, first display: \frac{\partial^2}{\partial x \partial y} should be \frac{\partial^2}{\partial y \partial x}.
  • p. 5, line 6 from bottom: \sin(\pi/2-2) should be \sin(\pi/2-z).  (Actually, for pedagogical reasons, it may be slightly better to use \pi/2+z throughout this example instead of \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 (on page 60): 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.
  • Page 181: In Lemma 7.1.4(c), a period is missing at the end of (\sum_{i=m}^n b_i).
  • p. 183: In the proof of Proposition 7.1.8, x should be replaced by  f(x) in every display of the proof in which it appears.
  • 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 (hardback) —

  • p. xii, bottom: “solidifed” –> “solidified”.
  • p. xiv, top: “to know how to to” –> “to know how to”.
  • p. 19.  In footnote 2, add: “In the converse direction, if we have n=m, then we may deduce n++=m++; this is the axiom of substitution (see Appendix A.7) applied to the operation ++.”
  • p. 24, after Definition 2.2.1: “defined n+m for every integer n” should be “defined n+m for every natural number n“.
  • p. 26, after Proposition 2.2.6:  “these notes” should be “this text”.
  • p. 28, Proposition 2.2.14: “and Let” should be “and let”.
  • p. 30, Lemma 2.3.3: “Natural numbers have no zero divisors” should read “Positive natural numbers have no zero divisors”.
  • p. 32, Definition 2.3.11: Add the remark “In particular, we define 0^0 to equal 1.”
  • p. 37, Example 3.1.10: “(why?)” should be “(why?))”.
  • p. 45: “8-m, where n is a…” should be “8-m, where m is a…”.  In Exercise 3.1.2, add Axiom 3.1 to the list of permitted axioms.  In Exercise 3.1.1: (3.1.4) should be Definition 3.1.4.
  • p. 50: In the first line, h(2n+3)=h(2n+2) should be h(2n+3)=2n+2, and N \backslash \{0\} should be {\bf N} \backslash \{0\}.
  • p. 55, Exercise 3.3.1: f \circ g and \tilde f \circ \tilde g should be g \circ f and \tilde g \circ \tilde f respectively.
  • p. 59: In Lemma 3.4.9, “Then the set … is a set” should read “Then there is a unique set of the form … .  That is to say, there is a set A such that for any Y, Y \in A if and only if Y is a subset of X.
  • p. 61: In Exercise 3.4.8, Axiom 3.1 should be added to the list of permitted axioms.
  • p. 64: In Example 3.5.9,  “(x_2,x_3) \in X_3” should be “(x_2,x_3) \in X_2 \times X_3“.
  • p. 70, 4th line of proof of Lemma 3.6.9: 1 \leq i \leq N should be 1 \leq i \leq n.  In the 6th line of proof of Proposition 3.6.8: Proposition 3.6.4 should be Lemma 3.6.9.  After Lemma 3.6.9, add the following remark: “Strictly speaking, the expression n-1 has not yet been defined.  For the purposes of this lemma, we temporarily define it to be the unique natural number m such that m++=n (which exists and is unique by Lemma 2.2.10).”
  • p. 81, before Lemma 4.2.3:  “product of a rational number” -> “product of two rational numbers”.
  • p. 84, before Definition 4.2.6: a space is missing between “Proposition 4.2.4” and “allows”.  Before this paragraph, add “In a similar spirit, we define subtraction on the rationals by the formula x-y := x + (-y), just as we did with the integers.”
  • p. 86: In Definition 4.3.2, “real numbers” should be “rational numbers”.  In definition 4.3.4, “be a rational number” should be added after “Let \varepsilon>0“.
  • p. 88: In Proposition 4.3.10(b), the hypothesis n>0 should be added.
  • p. 104, proof of Lemma 5.3.7; after invoking Proposition 4.3.7, add “(extended in the obvious manner to the \delta=0 case)”.
  • p. 105, after Proposition 5.3.10: \lim_{n \to\infty} a_n should be \hbox{LIM}_{n \to \infty} a_n.
  • p. 108, proof of Lemma 5.3.15: n \geq N should be n, m \geq N.  “This shows that |a_n-a_m| \leq \varepsilon” should read “This shows that |a_n^{-1}-a_m^{-1}| \leq \varepsilon“.
  • p. 115: In the hint for Exercise 5.4.8, add “or Corollary 5.4.10” after “use Proposition 5.4.9”.
  • p. 120: Add an additional exercise, Exercise 5.5.5:  “Establish an analogue of Proposition 5.4.14, in which “rational” is replaced by “irrational”.”
  • p. 124, Exercise 5.6.3: Add the hypothesis that x is non-zero (since the roots of 0 are not yet defined).
  • p. 126, proof of Proposition 6.1.4: Proposition 5.4.14 should be Proposition 5.4.12.
  • p. 134: In Definition 6.2.6(c) (and also on the first line of p. 135), E - \{-\infty\} should be E \backslash \{-\infty\}.
  • p. 135, Theorem 6.2.11(b), (c): Replace “Suppose that M” with “Suppose that M \in {\Bbb R}^*” (two occurrences). Exercise 6.2.2: Proposition 6.2.11 should be Theorem 6.2.11.
  • p.144: Cor. 6.4.14: line 4: ” .. for all n \geq M”  should be  ” .. for all n \geq m
  • p.146: proof of Theorem 6.4.18: Replace “from Corollary 6.1.17” here by “from Lemma 5.1.15 (or more precisely, the extension of that lemma to the real numbers, which is proven in exactly the same fashion)”.
  • p. 151, 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. 164, Definition 7.2.2: (S_N)_{n=m}^\infty should be (S_N)_{N=m}^\infty.
  • p. 169, 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?”.  After Corollary 7.3.2: “conditionally divergent” should be “not conditionally convergent”, similarly in Exercise 7.2.13.
  • p. 176: “absolutely divergent series” should be “series that is not absolutely convergent”.
  • p. 177, Theorem 7.5.1: “conditionally divergent” should be “not conditionally convergent”, and similarly “absolutely divergent” should be “not absolutely convergent”.  Similarly for Corollary 7.5.3 on page 179.
  • p. 186, Exercise 8.1.1: This exercise requires the axiom of choice, Axiom 8.1.  In Exercise 8.1.4. f(0), f(1), \ldots, f(n) should be f(0), f(1),\ldots,f(n-1).
  • p. 192, proof of Theorem 8.2.8: “absolutely divergent” should be “not absolutely convergent” (two occurrences).
  • p. 196, Remark 8.3.6: “Paul Cohen (1934-)” should now be “Paul Cohen (1934-2007)”.  :-(
  • p. 197, 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. 200, Exercise 8.4.1: y \in y should be y \in Y.
  • p. 206, Exercise 8.5.5: “f(x) \leq_Y f(x')” should be “$latex f(x)
  • p. 208, Exercise 8.5.19: Y := \{y \in Y': y < x \} should be Y = \{ y \in Y': y <' x \}.  In Exercise 8.5.20, the additional hypothesis “Assume that \Omega does not contain the empty set \emptyset” should be added.
  • p. 214, Lemma 9.1.21.  One needs the additional hypothesis “We assume that a<b.”
  • p. 220, Definition 9.3.6: “f is \varepsilon-close to L near x_0” should be “f, after restricting to E, is \varepsilon-close to L near x_0“.
  • p. 228, Proposition 9.4.7: change “three items” to “four items”, and add “(d): For every \varepsilon > 0, there exists a \delta > 0 such that |f(x)-f(x_0)| \leq \varepsilon for all x \in X with |x-x_0| \leq \delta.
  • p. 232, proof of Proposition 9.5.3: after “Proposition 9.4.7”, add “(applied to the restriction of f to the subdomain X \cap (x_0,+\infty))”.
  • p. 252, 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. 253, Definition 10.1.11: “For every x_0 \in X” should be “For every limit point x_0 \in X“.
  • p. 254, Remark 10.1.14: Leibnitz should be Leibniz (two occurrences).
  • p. 256, Exercise 10.1.1: “x_0 is also limit point of Y” should be “x_0 \in Y, and x_0 is also a limit point of Y“.
  • p. 257, Definition 10.2.1: x \in X should be x_0 \in X.
  • p. 262: In the proof of Theorem 10.4.2,”x_n = f^{-1}(y_0)” should be “x_n = f^{-1}(y_n)“.
  • p. 271, Remark 11.2.2: “constant on f” should be “constant on E“.
  • p. 290: In Exercise 11.6.5, add “For this exercise, you may use the second Fundamental Theorem of Calculus (Theorem 11.9.4); there is no circularity, because Corollary 11.6.5 is not used in the proof of that theorem.”
  • p. 290: In the proof of Proposition 11.7.1, in the third display, [0,1] should be |[0,1]|.
  • p. 299: In Exercise 11.9.1, the hint is misleading (it requires the mean value theorem for integrals rather than for derivatives, which is not covered in this text) and should be deleted.

— Errata to the third edition (hardback) —

  • General note: all references to “Analysis II” need to be renumbered to account for the new chapter numbering (basically, all chapter numbers need to be lowered by 11.)
  • Page 10, footnote: “f(0,0) := (0,0)” should be f(0,0) := 0“.
  • Page 15: In Section 2.1, “Guiseppe Peano” should be “Giuseppe Peano”.
  • Page 29: In the hint for Exercise 2.2.5, n < m_0 should be n \leq m_0.
  • Page 35: Definition 3.1.4 has to be given the status of an axiom (the axiom of extensionality) rather than a definition, changing all references to this definition accordingly.  This requires some changes to the text discussing this definition. Firstly, in the preceding paragraph, “define the notion of equality” will now be “seek to capture the notion of equality”, and “formalize this as a definition” should be “formalize this as an axiom”.  For the paragraph after Example 3.1.5, delete the first two sentences, and remove the word “Thus” from the third sentence.  Exercise 3.1.1 is now trivial and can be deleted.
  • Page 37: In Example 3.1.10, “so is singleton set” should be “the singleton set”; also, a right parenthesis is missing after (why?).  In Axiom 3.4, “elements consists” should be “elements consist”.
  • Page 52: In the first paragraph of Section 3.2, the appearances of the word “both” should be deleted.
  • Page 54: In Definition 3.3.17, the remark that a function is onto if f(X)=Y should be moved to the next section, because the image f(X) is not defined until that section.
  • Page 60: A space missing between “the” and “Zermelo” in Remark 3.4.12.
  • Page 62: In Example 3.3.22, “Axioms 2.2, 2.3, 2.4” should be “Lemma 2.2.10”.
  • Page 64: The justification that the product set \prod_{i=1}^n X_i given in Remark 3.5.8 is not quite correct if one is using the definition of an ordered n-tuple as defined in Exercise 3.5.2 (one has to restrict the range of the tuples to be surjective).  As the correct version of this remark is part of Exercise 3.5.2, the second sentence of this remark should be replaced with a reference to that exercise.
  • Page 67: In Exercise 3.5.12, a_N(n++) = f(n,a(n)) should be a_N(n++) = f(n,a_N(n)).
  • Page 68: In Example 3.6.2, there is a superfluous period before the parenthetical (also the period after the parenthetical should be inside).
  • Page 70: In the proof of Lemma 3.6.9, “Now define the function g: X - \{x\} to \{ i \in {\bf N}: 1 \leq i \leq n-1\}” should be “Now define the function g: X - \{x\} \to \{ i \in {\bf N}: 1 \leq i \leq n-1\}” .  In the 4th line of proof of Lemma 3.6.9: 1 \leq i \leq N should be 1 \leq i \leq n.
  • Page 72: In Exercise 3.6.8, the additional hypothesis that A is non-empty should be added.  Also, the word “then” may be deleted.
  • Page ???: In the footnote preceding Definition 4.2.1, add in the first sentence “… and a is non-zero.  Similarly, the identities a/a = 1 and 2*(a/a) = (2*a)/a cannot hold simultaneously if 0/0 is defined.”
  • Page 97: In Definition 5.1.6 and Definition 5.1.8, d(a_j,a_k) should be |a_j-a_k| (for consistency with later definitions).
  • Page 103: Near Proposition 5.3.3, “laws of equality” should be “axioms of equality”, and “law of substitution” should be “axiom of substitution”.
  • Page 112: In Definition 5.4.6, “if” should be “iff”.
  • Page 144: Below the proof of Proposition 6.4.12, a right parenthesis should be added after “(provided that L^+ and L^- are finite”.  Also, “(c) and (d)” should be “(d) and (e)”.
  • Page 150: In Example 6.6.3, 0.001 should be inserted between 1.001 and 1.0001.
  • Page 152: In Exercise 6.6.3, add the following note: “To ensure the existence and uniqueness of the minimum, one either needs to invoke the well ordering principle (which we have placed in Proposition 8.1.4, but whose proof does not rely on any material not already presented), or the least upper bound principle (Theorem 5.5.9).” Similarly for Exercise 6.6.5.
  • Page 153: In the proof of Lemma 6.7.1, the first equal sign in the display d(x^{q_n}, x^{q_m}) = x^M (x^{q_n-q_m}-1) \leq \dots should be a \leq sign.
  • Page 158: In Example 7.1.7, h(3)=c should be h(3)=a.
  • Page 160:  In Remark 7.1.10, all occurrences of f(x) here should be f(n).
  • Page 162: In the third to last display, the small parenthesis near the end of the first term on the RHS should be moved to the outside (also, this pair of parentheses should be made larger).
  • Page 167: In the proof of Proposition 7.2.12, “the sequence (-1)^n a_n” should be “the sequence ((-1)^n a_n)_{n=m}^\infty“; similarly for “the sequence S_n” and “the sequence a_N“.
  • Page 174: In the proof of Proposition 7.4.1, (S_N)_{n=0}^\infty and (T_M)_{m=0}^\infty should be (S_N)_{N=0}^\infty and (T_M)_{M=0}^\infty respectively.
  • Page 175: In the first sentence, m \leq N should be m \leq M.
  • Page 176: In the proof of Proposition 7.4.3, “\varepsilon-close to L” should be \varepsilon-close to L'” in the last paragraph.
  • Page 191: In Lemma 8.2.3, X should be assumed to be countable, rather than at most countable.
  • Page 193: In Lemma 8.2.7, the last sentence should read “Then the series \sum_{n \in A_+} a_n and \sum_{n \in A_-} a_n are not absolutely convergent.”
  • Page ???: Near the end of proof of Theorem 8.2.8, it would be (slightly) better to have \lim_{j \to \infty} \sum_{0 \leq i \leq j} a_n rather than \lim_{j \to \infty} \sum_{0 \leq i < j} a_n.
  • Page 202: In Exercise 8.4.3, “there exists an injection f: A \to B; in other words…” should be “there exists an injection f: A \to B with g \circ f: A \to A the identity map; in particular…”.  (This is needed in order to establish the converse part of the question.)
  • Page 209: In Exercise 8.5.16, “x,y \in P” should be “x,y \in X“.  In Exercise 8.5.18: A right parenthesis is missing after “… which contains Y“.  “Tthus” should be “Thus”.
  • Page 217: In Exercise 9.1.15, the hypothesis that E is non-empty should be added.
  • Page 226: In Example 9.3.21, all sequences here should start from n=1 rather than from n=0.
  • Page 295: In the last paragraph of Section 11.8, a right parenthesis should be added at the end of the penultimate sentence.
  • Page ???: In Example 12.7.3, “the substitution axiom” should read “the first form of the substitution axiom”.  Then, at the end of the example, add “One can also obtain the conclusion x=\sin(z^2) more directly by using the second form of the substitution axiom.”.

Note that the first edition paperback page numbers differ from the second (or third) edition hardback page numbers, which should be born in mind when applying the second edition errata to the first edition. (The section, theorem and exercise numbering, however, is mostly unchanged.)

Thanks to Adam, James Ameril, José Antonio Lara Benítez, Tai-Danae Bradley, Brian, Eduardo Buscicchio, Matheus Silva Costa, Gonzales Castillo Cristhian, Ck, Lorenzo Dragani, Evangelos Georgiadis, Elie Goudout, Ti Gong, Christian Gz., Ulrich Groh, Erik Koelink, Matthis Lehmkühler, Percy Li, Ming Li, Manoranjan Majji, Mercedes Mata, Pieter Naaijkens, Vineet Nair, Cristina Pereyra, David Radnell, Tim Reijnders, Pieter Roffelsen, Luke Rogers, Feras Saad, Gabriel Salmerón, Marc Schoolderman, Chaitanya Tappu, Kent Van Vels, Daan Wanrooy, Yandong Xiao, Luqing Ye, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.