Last updated Dec 5, 2022

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 prior to the corrected third edition may be found here.

— 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. In Exercise 1.1.11, should be .
- Page 15: In Proposition 1.2.15, should be (two occurrences).
- Page 16: In the first paragraph, the first parenthetical comment should be closed after “… and hence outside of .” In the second parenthetical comment, the period should be outside the parenthesis. “The point 0” should be “The point ” (two occurrences).
- Page 21: In Exercie 1.4.7 (b), should be .
- Page 22: In Definition 1.5.3, insert “for every ” before “there exists a ball” (in order to keep the empty metric space bounded). Also, add the requirement that be finite.
- Page 23: In Theorem 1.5.8, should be in the statement of the theorem (four occurrences). In Case 2 of the proof, should be . One should in fact split into three cases, , , and . For the last case, write “For this case we argue as in Case 2, but replacing the role of by (say) “. In the proof of Theorem 1.5.8, should be .
- Page 26: In Exercise 1.5.10, 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 should be .
- Page 30: In Exercise 2.1.7, should be .
- Page 33: Add an additional Exercise 2.2.12 after Exercise 2.2.11: “Let be the function defined by when , and when . Show that for every , but that 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 34: In Proposition 2.3.2, replace “Furthermore, ” with “Furthermore, if is non-empty”,
- Page 37: In Theorem 2.4.5, replace “Let be a subset…” with “Let be a non-empty subset…”.
- Page 38: In 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 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.
- Page 43: Exercise 2.5.8 is
**incorrect**(the space is sequentially compact) and should be deleted. - Page 44: In Exercise 2.5.14, add “Hausdorff” before “topological space”.
- Page 46: In Definition 3.1.1, the domain of should be rather than . Similarly for Proposition 3.1.5, Exercise 3.1.3, and Exercise 3.1.5. In Remark 3.1.2, should be .
- Page 47: In Proposition 3.1.5(c), all occurrences of should be .
- Page 48: In Exercise 3.1.1, add the hypothesis “Assume that is an adherent point of (or equivalently, that is not an
*isolated point*of )”. In Exercise 3.1.3, replace the last three sentences with “If is a topological space and 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 is not Hausdorff?”. - Page 52: In the last paragraph of the section, should be .
- Page 56: In 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 metric”, and restrict the definition of to the case when is non-empty, then add “When is empty, we instead define “; 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 in place of .
- Page 60: In Example 3.5.8, “ratio test” should be “root test”, and Theorem 7.5.1 should be “from Analysis I”. Also “ converges uniformly” should be “ converges uniformly”.
- Page 61: In the second to last display, the factor in front of should be deleted.
- Page 62: In Example 3.6.3, Lemma 7.3.3 should be “from Analysis I”.
- Page 64: At the end of the first paragraph, the period should be inside the parentheses.
- Page 76: In the first display of Example 4.1.5, should be .
- Page 77: In Remark 4.1.9, it is more appropriate to add “uniformly” after “assures us that the power series will converge”.
- Page 78: At the end of the Exercise 4.1.1, a right parenthesis should be added.
- Page 79: In Definition 4.2.4, add “with the property that every element of is a limit point of ” at the end of the first sentence. At the end of the second sentence, add “, in particular is also a function on .”
- Page 81: In Corollary 4.2.12, “ecah” should be “each”.
- 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. 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 in which the are replaced by . At the beginning of the exercise, “anaytic in ” should be “analytic at “.
- Page 86: In the last two displays, should be .
- Page 91: Before Definition 4.5.5, “exp is increasing” should be “exp is strictly increasing”.
- Page 92: At the end of Exercise 4.5.1, a right parenthesis should be added.
- Page 99: before the final paragraph, add “Inspired by Proposition 4.5.4, we shall use and interchangeably. It is also possible to define for complex and real , but we will not need to do so in this text.”
- 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 112: In Example 5.2.6, should be .
- Page 113: In 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.
- 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, should be . Expand the sentence “The composition of two linear transformations is again a linear transformation (Exercise 6.1.2).” to “The composition of two linear transformations is again a linear transformation (Exercise 6.1.2). It is customary in linear algebra to abbreviate such compositions of linear transformations by droppinng the symbol, thus .”
- Page 134: In Lemma 6.2.1, “, and ” should be “, and let be a limit point of “. In the previous display, should be .
- Page 135: In the first paragraph, the period should be inside the parenthesis. In Definition 6.2.2, should be a limit point of .
- 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. Expand the third display to “, and expand “From Lemma 6.3.5” to “From Lemma 6.3.5 (and Proposition 9.5.3 from Analysis I)”.
- Page 140: In the beginning of the proof of Theorem 6.3.8, should be , and similarly the sum on the RHS should be from to rather than from to . “Because each partial derivative … is continuous on ” should be “Because each partial derivative … exists on and is continuous at “.
- Page 141: The period in the last line (before “and so forth”) should be deleted.
- Page 142: At the end of the page, should be .
- Page 144: In Exercise 6.3.2, should be .
- Page 146: In the second paragraph, third sentence, the period should be inside the parenthesis.
- Page 148: In the proof of Clairaut’s theorem, should be .
- Page 151: In Exercise 6.6.1, the range of should be rather than .
- Page 153: In the second paragraph of the proof of Theorem 6.7.2, “ is
*not*invertible” should be “ is*not*invertible”. - Page 154: In the last text line, can be for clarity.
- Page 155: In 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.”
- Page 156: should be . The definition of should be rather than (and the later reference to can be replaced just by ).
- Page 157: In the final paragraph of Section 6.7, “differentiable at ” should be “differentiable at “. 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 if necessary (while still having , $f(x_0) \in V$ of course), the derivative map is invertible for all , and that the inverse map is differentiable at every point of with for all . Finally, show that is continuously differentiable on .”
- 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 if necessary , that the function becomes continuously differentiable on all of , and the equation (6.1) holds at all points of .”
- Page 163: after “if and are disjoint”, add “, and more generally, that when are disjoint”.
- Page 164: In the first paragraph of Section 7.1, should be .
- Page 165: Superfluous 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 of non-negative quantities is equal to if the sum is not absolutely convergent.”
- Page 169: After (7.1), should be .
- Page 170: In the first paragraph “For all other values if ” should be “For all other values of “.
- Page 172: should be (three occurrences). In Example 7.2.9, “each rational number ” should be “each rational number “.
- Page 173: In Exercise 7.2.2, final sentence: period should be inside parentheses. Also, add “Here we adopt the convention that is infinite for any and vanishes for .” In Example 7.2.12, “countable additivity” should be “countable sub-additivity”.
- Page 174: In the penultimate paragraph, “identical or distinct” should be “identical or disjoint”, and should be . Also, “coset of ” should be “coset of “; in the next paragraph, “the rationals ” should be “the rationals “. In Exercise 7.2.5, “Q1” should be “Exercise 7.2.3”.
- Page 175: In the second paragraph, “constrution” should be “construction”. After the third paragraph, add “Note also that the translates for are all disjoint. For, if there were two distinct with intersecting , then there would be such that . But then and thus , which implies , contradicting the hypothesis.”
- Page 176: In the proof of Proposition 7.3.3, “cardinality 3n” should be “cardinality “.
- Page 178: In Lemma 7.4.5, “and any set ” should be “then for any set “.
- Page 180: In 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).
- Page 181: “… on our wish list is (a)” should be “… on our wish list is (i)”.
- 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 188: In Lemma 8.1.5 the function should take values in rather than (and then the requirement that be non-negative can be deleted).
- Page 189: In 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 rather than , and “are a finite number” should be “be a finite number”. In Example 8.1.7, “the integral” should be “the interval”.
- Page 190: In the final display in the proof of Lemma 8.1.9, an equals sign should be inserted to the left of the final line.
- Page 194: In Theorem 8.2.9, should take values in rather than .
- Page 196: Before 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 are measurable”. In the first paragraph, all instances of should be .
- Page 197: After the third display. Proposition 8.1.9(b) should be Proposition 8.1.10(bd).
- Page 199: Exercise 8.2.4 should be moved to Section 8.3 (as it uses the absolutely convergent integral).
- Page 200: In the hint to Exercise 8.2.10, the “for all ” should be moved inside the set builder notation , thus using instead.
- Page 201: Before Definition 8.3.2, when Corollary 7.5.6 is invoked, add “(which can be extended to functions taking values in without difficulty)”.
- Page 202: In the start of the proof of Theorem 8.3.4, add “If was infinite on a set of positive measure then would not be absolutely integrable; thus the set where is infinite has zero measure. We may delete this set from (this does not affect any of the integrals) and thus assume without loss of generality that is finite for every , which implies the same assertion for the .
- Page 204: In the second display, should be instead.
- Page 205: In Proposition 8.4.1, add the hypothesis that is bounded.
- Page 206: In the last paragraph, last sentence, the period should be inside the parentheses. In the last two displays, should be .
- Page 207: In the third paragraph, “Secondly, we could fix” should be “Thirdly, we could fix”.
- Page 208: In the last paragraph, Lemma 8.1.4 should be Lemma 8.1.5.

— Errata for the fourth edition —

- Page 4: In Example 1.1.9, should be .
- Page 14: In Proposition 1.2.15(c), delete the word “then”.
- Page 32: In Corollary 2.2.3(b), the modifiers “at ” should be deleted. In Corollary 2.2.3(a), the modifier “at ” should be added after “ is continuous”.
- Page 35: in the proof of Theorem 2.3.5, one should first dispose of the case separately (in which all functions are both continuous and uniformly continuous for vacuous reasons).
- Page 42: In Exercise 2.5.4, when referring to the trivial topology being non-Hausdorff, add the hypothesis that the space contains at least two points.
- Page 48: In Exercise 3.1.5, add the hypothesis .
- Page 53: In Exercise 3.2.2(c), Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I.
- Page 55: In Remark 3.3.7, “it only works” should be “they only work”. In Exercise 3.3.2, replace “cannot be used” with “cannot immediately be used” in the hint.
- Page 84 onwards: any appearance of colons in limits, such as , should be replaced with semicolons for consistency.
- Page 98: In the final display in Lemma 4.6.14, the parentheses should be matched in size.
- Page 107: “Napoleons” should be “Napoleon’s”.
- Page ???: In Lemma 5.3.5, should be .
- Page 118?: in the proof of Lemma 5.4.6, the reference to Lemma 7.3.3 should instead be Exercise 7.3.2 from Analysis I.
- Page ???: In Exercise 6.1.4, give the definition of the norm (repeating the definition given in Definition 6.2.2).
- Page 150: In exercise 6.6.7, Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I.
- Page 152: In the second paragraph of Section 17.7, “ is not invertible” should be “ is not invertible”.
- Page 166: “open intervals” should be “open bounded intervals”.
- Page 167: “for every box ” should be “for every box “.
- Page 170: In Example 7.2.11, “ has outer measure” should be “ has outer measure”.
- Page 182: In Corollary 7.5.7, “then so is” should be “then so are”.
- Page 187: In Example 8.1.7, “simple integral” should be “Lebesgue integral”.
- Page 204: before the third to last display in the proof of Lemma 8.3.6, “but” should be capitalised. In the second display, the upper bound should instead be a lower bound .
- Page 206: In Exercise 8.3.3, should be .
- Page 207: In Theorem 8.5.1, “Then there exists” should be “Then there exist”.
- Page 201: In the fourth display, a is missing in the third integral.

Caution: the page numbering is not consistent across editions. Starting 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, Dingjun Bian, Petrus Bianchi, Philip Blagoveschensky, Carlos, cebismellim, William Deng, EO, Florian, Aditya Ghosh, Gökhan Güçlü, Yaver Gulusoy, Kyle Hambrook, Minyoung Jeong, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Brett Lane, Matthis Lehmkühler, Zijun Liu, Rami Luisto, Jason M., Matthew, Manoranjan Majji, Geoff Mess, Guillaume OlikierJorge Peña-Vélez, Cristina Pereyra, Issa Rice, Frédéric Santos, SkysubO, Rafał Szlendak, Winston Tsai, Kent Van Vels, Andrew Verras, Murtaza Wani, Xueping, Sam Xu, Zelin, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

## 418 comments

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18 January, 2022 at 1:16 am

FredDear Prof. Tao,

A possible typo on top of page 167:

“if we wish for every box ” might be clearer as “for every box ” or simply “for all “.

[Correction added, thanks – T.]27 January, 2022 at 7:19 am

Frédéric SantosDear Prof. Tao,

Some possible other minor typos for Chapter 7:

– Page 172: in Example 7.2.9, “each rational number ” should be “each rational number “.

– Page 174: in Exercise 7.2.5, the Hint makes reference to an unknown “Q1”.

– Page 176: in the proof of Proposition 7.3.3, “3n” should be in math mode.

– Page 181: on top of the page, I think that “the final property … on our wish list is (a)” should be “… is (i)”.

[Errata added, thanks – T.]1 February, 2022 at 9:02 am

Frédéric SantosDear Prof Tao,

Another minor detail of convention / notation.

Page 84, and several other pages in Section 4.3: starting with Theorem 4.3.1, the limits are noted with a colon () instead of the previous convention that consistently uses a semicolon in the whole text (). All instances of “:” might be replaced by “;” in this context.

[Correction added, thanks – T.]8 February, 2022 at 12:36 am

Frédéric SantosDear Prof. Tao,

Here are some other minor typos or possible minor improvements:

– Page 29 (Corollary 2.1.7), and also page 48 (Exercise 3.1.5): an additional hypothesis that might be added.

– Pages 30-31, Corollary 2.2.3: I think that “at ” should be added at the end of statement (a), and should instead be removed from the two sentences in statement (b).

– Page 98, last line of the page: the inner parentheses might be enlarged in the right-hand side of the display.

– Page 166: just for an additional precision, in the sentence just after Definition 7.2.1, we could say “… are the same as open bounded intervals” instead of “… as open intervals”.

– Page 204: at the beginning of a line (~ middle of the page), “but” might be “But” (or the period just before might be turned into a comma).

– Page 205, Exercise 8.3.3: at the end of the second line, should be .

– Page 210, in a display just after “and hence,”: a “” might be missing just before the closing parenthesis, in the second integral.

[Corrections added, thanks – T.]10 April, 2022 at 1:51 pm

AnonymousPage 204 : Shouldn’t it be >= A-1/n ?

[Erratum added, thanks – T.]7 July, 2022 at 11:51 am

shinkenjoeHello,

I didn’t see it in the comments and errata. I would drop the support of f on [0,1] for 3.8.18 p. 71 in 3rd ed. I think 3.8.16 was only introduced to be able to drop it.

Byes.

[Actually the arrangement is as intended. -T]7 July, 2022 at 12:06 pm

folkerttI can not really comment on this, as my eyesight is not good enough to check such a what-must-be-a-simple statement to check and so easy to

oversee. Avoiding making such a mistake takes an uncommon amount of rigor, a state of perfection only observed at death. I know shinkenjoe to be a good man, and he has the best interests of the world at heart, same as tt. I hope this can be fixed someday and not become a trigger event for www if you can count the number of w’s. But who can still count to 3. 1+2=3. It was beaten in me. Some people

have opinions about this and say 1+1=0. In that case, he would have killed me, and people would have complimented him. That was then

and now is now, and tomorrow could have been toast.

Prosit. Mea Culpa, at 4 pm.

12 October, 2022 at 2:27 am

PingoHi prof Tao!

A question about Exercise 2.4.2. This exercise can be easily solved if we use Theorem 2.4.6. But since this exercise is placed before Exercise 2.4.4, this suggests there exists a way to solve the exercise without using Theorem 2.4.6 (but it seems significantly more difficult). Are we allowed to use Theorem 2.4.6 here? (If the answer is yes, it might be better to place Exercise 2.4.2 after Exercise 2.4.4?)

Thanks!

[Fair enough; I have added an erratum regarding the suggested move. -T.]4 November, 2022 at 1:34 am

MatthewDear professor Tao,

I think there is a minor typo regarding the proof of Theorem 2.3.5 on page 35. One should break into cases depending on whether or . Since if , then the initial collection of balls that one constructs would also be empty. Then the finite number of points one finds are just no points at all. And then the positive number cannot be defined.

[Erratum added, thanks – T.]2 December, 2022 at 3:21 am

Guillaume OlikierDear Prof. Tao,

I believe that there is a typo on page 152 of the third edition: in the first sentence of the second paragraph of Section 6.7, I think that “when f’ is not invertible” should be replaced by “when f is not invertible”. Is it correct?

Thank you in advance for your answer.

Best regards,

Guillaume Olikier

PhD Student at UCLouvain (Belgium)

[Correction added, thanks – T.]3 December, 2022 at 2:10 am

Frédéric SantosHi Prof. Tao,

In Exercise 6.1.4, we use the notion of the norm of a vector , but this notion is not generally defined in the previous chapters of the book. (We do have a notion of inner product and norm for periodic functions in the previous chapter, but we don’t have any norm on until Definition 6.2.2). A hint might be added at the end of Exercise 6.1.4 to address this issue (e.g., “see Definition 6.2.2”).

Also, I think that in Lemma 5.3.5, should be , for clarity (still because we only have definitions for particular norms, and no general definition of normed vector spaces).

All the best,

F.

[Correction added, thanks – T.]3 December, 2022 at 12:46 pm

AnonymousIf the norm is not specified, it usually mean the Euclidean norm.

4 December, 2022 at 1:45 am

Frédéric SantosI may be wrong, of course, but it might be a little trickier than that: if the *distance* is not specified, it always means the euclidean distance, indeed. But the concept of norm on in itself is not introduced before Definition 6.2.2 (and is called length rather than norm). I think that the notation for is never used in the book before Exercise 6.1.4, thus this suggestion.

27 December, 2022 at 1:14 pm

AnonymousDear Prof. Tao

Page 193, shouldn’t be the integral on \omega of f_restricted to \omega’ ?

[No; the equation is correct as written. (An integral on can only integrate functions defined on , and is only defined on the smaller set .) -T.]