Last updated July 17 2021

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: should be .
- p. 393, example 12.1.9: should be .
- 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: should be .
- p. 416, line 11: “” should be ““.
- p. 419, line -2: In Exercise 12.5.15, = should be . Also, “that by counterexample” should be “by counterexample that”
- p. 426, Exercise 13.2.9: should be 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 and .”
- 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: “” should be ““.
- p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
- p. 459: In line 11, should be .
- 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, should be , 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, should be .
- 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, should be .
- p. 482: In Exercise 15.2.5, the on the right-hand side should be .
- p. 486: In second and third display, y should be in rather than .
- p. 493: In Exercise 15.5.4, should be .
- p. 501: In Theorem 17.7.2, “if is not invertible” should be “if 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 and not just in ; insert the line “By working with one component of at a time, we may assume ” as the first line of the proof. Also, should be .
- p. 557: In the second display, should be .
- p. 560: In Exercise 17.6.1, add the hypothesis “and is continuous” before “, show that is a strict contraction”.
- p. 561: In Exercise 17.6.3, change “which is a strict contraction” to “such that for all distinct in “. In Exercise 17.6.8, should be .
- p. 562: In Theorem 17.7.2, should be .
- p. 565, line -7: should be rather than .
- p. 570, first display: all partial derivatives should have a – sign (not just the first one). Last paragraph: “Thus lies in W” should be “Thus lies in U”.
- p. 571, second display: add “” at the end.
- p. 584, Corollary 18.2.7: “” should be ““.
- p. 599, Definition 18.5.9: should be .
- p. 600: In Lemma 18.5.10, should be . In the second and fourth lines of the proof of this lemma, should be .
- p. 616-617, Exercise 19.2.10: should be 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 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 “ should be 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 such that the “interval” ” should be replaced with “there exists a set which is an interval for some , a ray for some , the ray for some , or the whole space , 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 that has a maximal element , and such that the initial segments are countable for all . (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 .) If we give the order topology (Exercise 13.5.5), show that is compact; however, show that not every sequence has a convergent subsequence.”
- p. 395: In Proposition 14.1.5(d), add “Furthermore, if , then .”
- p. 396: In Exercise 14.1.5, should be , and should be .
- 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 ” should be “for some “.
- p. 431: In Exercise 15.2.8(e), “” should be “. In Exercise 15.2.8(d), should be .
- p. 433 (proof of Theorem 15.3.1): in the third display and in the next line should be

and respectively. - p. 452: In Exercise 15.7.2, should be . In Exercise 15.7.6, “complex real number” should be “complex number”.
- p. 473: In Exercise 16.5.4, should be .
- p. 477: In Example 17.1.7, should be .
- p. 486: In Definition 17.3.7, should be , and should be .
- 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, should equal rather than .
- p. 499, proof of Lemma 17.6.6: After “ does indeed map to itself.”, add “The same argument shows that for a sufficiently small , maps the closed ball to itself. After “ is a strict contraction”, add “on , and hence on the complete space “.
- p.502, proof of Theorem 17.7.2: “” should be ““.
- p. 505, Section 17.8: should be . In the second paragraph, the function should be (for better compatibility with the discussion of the implicit function theorem).
- p.508, proof of Theorem 17.8.1, “U is open and contains ” should be “U is open and contains “.
- p. 515: In the display before Definition 18.2.4, should be . In Definition 18.2.4, should be .
- p. 520: In Example 18.2.9, should be in the display.
- p. 528, proof of Lemma 18.4.8: On the second line, “let be any other measurable set” should be “let 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 and y=0, equals -1 when 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 21: In Exercise 1.4.8(c), add “(This part of the exercise requires the axiom of choice.)”.
- Page 22: In Definition 1.5.3, add “We call bounded if is bounded”.
- Page 32: In Corollary 2.2.3(b), “ is also continuous at ” should be “ is also continuous”, and similarly for “ are also continuous at “.
- Page 42: In Exercise 2.5.5, should be . In Exercise 2.5.13, “topological space” should be “Hausdorff topological space”. In Exercise 2.5.4, after “the trivial topology is not Hausdorff” add “if the space contains at least two elements”.
- Page 43: In Exercise 2.5.6, insert “and” between “if ” and “ is any…”.
- Page 48: In Exercise 3.1.5, should be .
- Page 71: In Corollaries 3.8.18 and 3.8.19, “supported on ” 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 of complex numbers is identical (as a metric space) to the Euclidean plane , since the complex distance between two complex numbers is exactly the same as the Euclidean distance between these points. Thus, every metric property that satisfies is also obeyed by ; for instance, 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 ” should be “continuous at “.
- 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 ‘s should be ‘s.
- Page 161: “if some other derivative is zero” should be “if some other derivative is non-zero”.
- Page 176: In Proposition 7.3.3, should be (two occurrences).
- Page 178: In Corollary 7.4.7, should be .

— 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 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”.
- 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 59: Before Definition 4.5.5, “exp is increasing” should be “exp is strictly increasing”.
- 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 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 .
- 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).
- 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 “.
- 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 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 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.

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, Dingjun Bian, Petrus Bianchi, Philip Blagoveschensky, Carlos, cebismellim, 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., Manoranjan Majji, Geoff Mess, Jorge Peña-Vélez, Cristina Pereyra, Issa Rice, 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.

## 381 comments

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22 October, 2020 at 7:08 pm

Winston TsaiIn Exercise 2.5.4, I think the trivial topology being not Hausdorff requires X to have more than one element.

[Erratum added, thanks – T.]23 October, 2020 at 2:09 am

Winston TsaiI think there is a typo in the fifth line of Exercise 2.5.6.

[Correction added, thanks – T.]23 October, 2020 at 12:20 pm

AnonymousAlso in Exercise 2.5.6, if the goal is to show that the cofinite topology cannot be obtained from a metric, could one simply use the fact that metric spaces are Hausdorff?

[That would work – T.]3 November, 2020 at 12:08 am

AnonymousIn the paragraph after Example 3.2.9, I think f|_Y should be f|_E.

[Erratum added -T.]3 November, 2020 at 3:34 pm

AnonymousIn Definition 3.4.2, if X is empty then I think the metric outputs .

[Good point; a special convention is needed in this case. -T]4 November, 2020 at 2:40 pm

AnonymousSimilarly for Definition 3.5.5 and Example 3.5.6.

[Erratum updated – T.]5 November, 2020 at 4:34 pm

WinstonIn Definition 4.2.4 where you define k-times differentiability, it is implicitly assumed that F^(k) has domain E. Doesn’t this require every point in E to be a limit point?

This also leads me to another point. I think the “derivative as a function” has never been defined (in either Analysis I or II). Only the value of the derivative of a function at a point has been defined. That is, let F be a function from E to R. Then the function F’ has not been defined, only F'(x_0) is defined for limit points x_0 of E for which this value exists.

Given your definition of differentiability on a domain (Definition 10.1.11), I’ve been working with the ad-hoc definition that F’ is just the function defined on the set of limit points of E for which the derivative exists.

Also, I still feel that “F is differentiable on S” should mean that the derivative exists at every point of S (and hence every point of S is a limit point of E), so we can write F'(x) for any x in S. Then when one says “F is differentiable” it means that F is differentiable on E, the domain of F.

See my comment on March 13, 2020 on the Analysis I page.

[Good point; I have now added the restriction that every point of is a limit point to the definition. -T.]5 November, 2020 at 8:42 pm

AnonymousI apologize if this seems amateurish, I’m just trying to organize my thoughts.

Let be a function.

We define the derivative function , where is set of limit points of for which the derivative exists.

We say that is differentiable on iff exists for all (so ).

The meaning of the original Definition 10.1.11 can always be recovered by letting be the set of limit points of .

If is differentiable on then we just say that is differentiable.

With these modifications I think Definition 4.2.4 can remain unchanged, although one could extend the definition to “k-times differentiable on ” by using the phrase “on ” in the appropriate spots. The definition of “kth derivative of ” remains unchanged (this is where we need a definition of a derivative function), but when is k-times differentiable on then we could define the “k-th derivative of on ” to be . This notion of k-times differentiable on a subset is used in Proposition 4.2.6.

Then also I think my two previous errata for Page 253 and Page 297 of Analysis I would also be reverted.

Would these definitions be worth making/modifying?

6 November, 2020 at 1:38 am

AnonymousActually, defining “k-th derivative of f on S” is pointless. To say that f is k-times differentiable on S is to say that S is a subset of the domain of f^(k). The function f^(k) will always exist, it’s just that it will often be the empty function.

14 November, 2020 at 12:31 am

WinstonIn Example 4.1.5, for the first fraction, I think the ‘n’ should be outside the parentheses.

[Correction added -T]17 November, 2020 at 1:15 am

WinstonIn Corollary 4.2.12, ‘each’ is mispelled.

[Correction added – T.]30 December, 2020 at 9:05 am

AnonymousThe irony…

11 December, 2020 at 6:15 pm

AnonymousIn Example 3.5.8, I think it should be that the series \sum f_n converges uniformly, not that f_n converges uniformly.

[Erratum added, thanks – T.]11 December, 2020 at 8:25 pm

AnonymousIn Definition 3.8.6(c) is the absolute value sign on f(x) redundant?

[Technically yes; however sometimes one can consider more general approximations to the identity in which the non-negativity hypothesis in (a) is omitted, in which case it is useful to retain the absolute value sign in (c). -T]13 December, 2020 at 11:00 pm

AnonymousIn Exercise 4.2.8, “real analytic in a” should be “real analytic at a”

[Erratum added – T.]19 December, 2020 at 1:27 pm

Kyle HambrookOn page 163, you give a set of reasonable properties for a measure defined on all subsets of R^n to obey: measure of unit cube is 1; finitely additive; monotone; translation invariant. The you write “Remarkably, it turns out that such a measure does not exist; one cannot assign a non-negative number to every subset of R^n which has the above properties.” This is not true. It is true if n >= 3 or if “finitely additive” is replaced by “countably additive.” But, if n=1 or n=2, there is a finitely-additive translation invariant extension of Lebesgue measure defined on all subsets of R^n. Original Reference: S. Banach, Sur le probleme de la mesure, Fund. Math. 4 (1923), 7-33.

[Erratum added, thanks – T.]20 December, 2020 at 1:26 pm

Kyle HambrookNot to be too picky, but my last name is “Hambrook”, not “Handbrook” as you wrote in the “thanks for corrections” section of this post.

[Corrected, thanks – T.]22 December, 2020 at 12:05 am

AnonymousIn Definition 6.2.2, the definition of derivative, should x_0 be a limit point of E?

[Erratum added – T.]24 December, 2020 at 8:07 pm

AnonymousIn Analysis 1 we only defined summation on countable sets J if there is a bijection from N to J which makes the corresponding infinite series absolutely convergent. In the definition of outer measure (Definition 7.2.4) and the subsequent material, should we take the sum of volumes to be +infinity if the sum is not absolutely convergent?

[Yes; an erratum has now been added to this effect. -T]28 December, 2020 at 7:18 pm

AnonymousIn Example 8.1.7, ‘integral’ should be ‘interval’.

[Corrected added, thanks – T.]29 December, 2020 at 1:54 am

AnonymousIn Exercise 8.2.4, I think we have not yet defined the integral for functions with negative values. Should we use Definition 8.3.2?

[Good point; erratum added. -T]29 December, 2020 at 4:56 pm

AnonymousIn Proposition 8.3.3, should f and g take values in the extended reals?

31 December, 2020 at 11:59 am

Terence TaoIn principle yes, but a technical difficulty arises in Proposition 8.3.3(b) in this case because might not always be everywhere defined. One can fix this by working with functions that are only defined almost everywhere rather than everywhere (and extending the Lebesgue integral to such absolutely integrable almost everywhere defined functions, taking advantage of Proposition 8.3.3(d)), and changing all the hypotheses in Proposition 8.3.3 to only be applied almost everywhere. This is done in my book on measure theory but for this text I am only giving the briefest introduction to the Lebesgue integral and chose not to dwell on these subtleties.

5 February, 2021 at 9:31 am

AnonymousIn Proposition 3.3.3, can one drop the assumption of completeness of ? I can’t find a counterexample.

[Yes, you are right; I’ve added an erratum to this effect. -T]5 February, 2021 at 2:51 pm

AnonymousOne can show that this is a Cauchy sequence using the assumptions that for each , exists and converges to uniformly.

Without completeness of , how can one show that the limit of the sequence exists?

If one drops the completeness of , does one needs to assume that exists (instead of a conclusion)?

5 February, 2021 at 4:14 pm

Terence TaoAh, that is correct, one runs into issues without completeness after all. A counterexample is , , , , , .

18 May, 2021 at 5:47 pm

samDear Professor Tao:

For Exercise 1.4.8(c): Given a Cauchy sequence of formal limits in the complete metric space, I was trying to show that it converges to the sequence of diagonal elements, but it did not seem to work. Does one have to use the axiom of choice for this part ? I’d be appreciated if you can provide some further hints.

[Yes; see errata for the third edition. -T]26 May, 2021 at 10:21 am

N.Dear Professor Tao,

in the third ed., p. 14, Prop. 1.2.15 g

for the intersection $\cap_{\alpha\in I}{F_\alpha}$ the index set $I$ should be assumed to be non-empty.

[This is not necessary, as by definition the intersection is when is empty. -T]7 June, 2021 at 1:57 pm

N.Thanks for the reply.

I know, that one can define the intersection as $X$ when $I$ is empty.

But in Analysis I, third ed., p. 60 you defined the intersection only for non-empty index sets:

“Note that if $I$ was empty, then

$\bigcup_{\alpha\in I}{A_\alpha}$ would automatically also be empty (why?).

We can similarly form intersections of families of sets, as long as the

index set is non-empty. More specifically, given any non-empty set $I$,

and given an assignment of a set $A_\alpha$ to each $\alpha\in I$, we can define the intersection $\bigcap_{\alpha\in I}{A_\alpha}$ by first choosing some element $\beta$ of $I$ (which we

can do since $I$ is non-empty), and setting

$\bigcap_{\alpha\in I}:=\{x\in A_\beta: x\in A_\alpha\forall\alpha\in I\}$,

which is a set by the axiom of specification.”

So I would recommend to add a remark somewhere, how the intersection should be defined in the case of $I=\emptyset$.

One can find a good explanation here:

https://en.wikipedia.org/wiki/Intersection_(set_theory)#Nullary_intersection

27 May, 2021 at 3:09 am

N.Dear Professor Tao,

in the third ed., p. 21, Ex. 1.4.8. (b)

you define a metric with codomain $\mathbb{R}^+$.

In Analysis 1, Ex. 5.5.3, you defined this set as the set of all positive real numbers.

Shoudn’t the codomain include 0?

Same problem: Analysis 2, Def. 3.4.2

[Errata added, thanks – T.]28 May, 2021 at 5:31 pm

samDear professor Tao:

Thank you for your reply to my question above on the Axiom of Choice and Exercise 1.4.8(c). However, I wonder if one can circumvent the use of Choice as follows: Given a Cauchy sequence in , the term corresponds to a Cauchy sequence in X, which permits some integer after which the terms are of distance at most apart, take the minimum of such integers and denote it by (Assuming to be increasing). We then consider the diagonal sequence formed by picking the term of the member of the original sequence in . May I know what is the issue with this argument?

7 June, 2021 at 12:57 pm

Terence TaoTo make this argument rigorous one would have to select a representative Cauchy sequence attached to each formal limit , and this is where the axiom of (countable) choice needs to be invoked.

9 June, 2021 at 9:39 am

N.Dear Professor Tao,

in Ex. 2.4.7 one is asked to prove that every path-connected subset E of X is connected.

The empty set is path-connected (vacuous truth), but in Def. 2.4.1 you declare the empty set as being special (neither connected nor disconnected.

[Erratum added to remove the empty case – T.]17 June, 2021 at 6:50 am

William DengDear Prof. Tao

I’m wondering about the footnote on the page containing Proposition where you talk about the “automobile” and “pedestrian” metrics. While I understand that given two points and in the city (modelled as a subset of ), the automobile distance between and is generally less than the pedestrian distance between and , for the purposes of discussing convergence of sequences with respect to different metric spaces (which is what the footnote refers to), I think any sequence of points converging in the automobile metric (so that the driving time goes to zero) will also converge in the pedestrian metric (so that the walking time goes to zero), and vice versa (it’s just that the “rate” of the convergence might be different).

19 June, 2021 at 10:18 am

Terence TaoThe example here was explicitly “whimsical” in nature and so should be interpreted loosely. For instance, imagine a person walking towards another person separated by some low fence or other barrier, until they come in contact. Informally this is representing convergence in the pedestrian metric, but if one used cars instead of people then one would not have convergence in the automobile metric if the automobile cannot (legally) cross the fence.

22 June, 2021 at 2:09 pm

AnonymousIn Stein-Shakarchi’s Fourier Analysis, The series is given as an example that is not the Fourier series of a Riemann integrable function. They argue by assuming it is the Fourier series for and the Abel mean at is

They write that (1) tends to infinity as tends to because diverges and this contradicts .

1. But what they are saying is , not . How can one justify exchange of the limit and the sum?

2. If one relaxes the Riemann integrable condition to Lebesgue integrable, can one find such ?

3. Given a sequence , are there well-known results for telling whether it is the Fourier coefficients of some function ?

23 June, 2021 at 2:14 am

ugrohRegarding your question 3.: There is a paper by Carathéodory from 1918. It is written in german (of course) and you find it in Mathematische Zeitschrift, 1918 on 309 – 320. There is a reference (footnote 1) answering your question (I hope).

23 June, 2021 at 10:07 am

AnonymousFor question 1, since , thus , so is dominated by , apply the dominated convergence theorem, we can interchange limit and summation.

23 June, 2021 at 10:25 am

AnonymousActually, never mind. The explanation above is not valid. I don’t think we are exchanging limit and sum here. We just need to find a value of such that the summation diverges.

26 June, 2021 at 6:01 am

N.Dear Professor Tao,

in the third ed., p. 77 (Theorem 4.1.6. (d)) you write:

… converges uniformly to f ‘ on the interval [a-r,a+r].

It should rather be:

… converges uniformly to f ‘(x) on the interval [a-r,a+r].

26 June, 2021 at 6:09 am

N.Dear Professor Tao,

in the third ed., p. 82 – Ex. 4.2.6 – one is asked to show that every polynomial P(x) of one variable is real analytic on the real numbers.

In Def. 4.2.1 you define real analytic functions.

A polynomial is an algebraic object, but not a function.

It only induces a function, as you can define a polynomial function by evaluating the polynomial.

So it should rather be:

Using Ex. 4.2.5, show that every polynomial function of one variable is real analytic on R.

[In these texts we often “abuse notation” by using a formal expression, such as a polynomial, to refer to the function it induces; see e.g., the discussion after Example 3.3.3 in Analysis I. -T]15 July, 2021 at 10:21 am

William DengDear Prof. Tao, shortly after stating Theorem , you say that “However, a version of the Heine-Borel theorem is available if one is willing to replace closedness with the stronger notion of completeness…”, but actually, I think intrinsic closedness is equally strong as completeness (i.e. from Proposition , we see that every complete metric space is also intrinsically closed, while combining with Exercise reveals that every intrinsically closed metric space is also complete).

[Completeness is indeed equivalent to being intrinsically closed, but this is not what being closed refers to by default. -T]15 July, 2021 at 6:25 pm

William DengA suggestion: for clarity, in Definition , replace “” by “” or “” or “finite ” or something to that effect (basically, you probably don’t intend to be a legitimate value for , otherwise for instance one can show using Definition that every metric space is bounded which would make the concept rather boring).

[The requirement that be finite will be added to Definition 1.5.3. -T.]16 July, 2021 at 7:07 am

William DengIs it possible to weaken the hypotheses in Corollary as follows: only assume to be compact and weaken the assumption of compactness on the other sets to closedness (in )? This is a strictly weaker assumption because although one can deduce from the compactness, hence boundedness of along with the totally ordered nature of the sets that the other sets are also closed (in ) and bounded, we saw previously that the full Heine-Borel theorem is not necessarily true for an arbitrary metric space.

16 July, 2021 at 8:12 am

William DengActually never mind, that would not be strictly weaker, rather it would just be an equivalent formulation, thanks to Theorem (so essentially, it seems that the full Heine-Borel theorem can be recovered when working inside a compact metric space).

17 July, 2021 at 1:04 pm

SimonI believe one of the corrections has a small typo. The correction of the beginning of the proof of Theorem 6.3.8 on page 140 has $n$ and $m$ swapped, i.e. it should say “[…] the sum on the RHS should be from $1$ to $n$ rather than from $1$ to $m$.”

[Corrected, thanks – T.]22 July, 2021 at 12:37 am

KevinDear prof. Tao, I believe the statement “Furthermore, if $x_0 \in E$, then $f(x_0) = L$” in Proposition 3.1.5 is wrong.

Consider the function $f : [0, 1] \to {\bf R}$ where $f(0.5) = 100$ and $f(x) = x$ for every $x \neq 0.5$.

Even if we can construct the function $g$, we cannot derived the continuous property for $f$ at $x_0$.