Last updated Jul 8, 2024
Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006. Fourth edition, 2022. Springer Fourth edition, 2022
Hardcover, 236 pages.ISBN 81-85931-62-3 (first edition), 978-981-19-7284-3 (Springer fourth 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 10: Expand Definition 1.2.9 to also define the notion of a limit point; similarly for Definition 2.5.6 on page 34.
- Page 12: In Exercise 1.2.2, “Lemma 8.4.5” should be “Lemma 8.4.5 from Analysis I”. In Proposition 1.2.15(c), delete the word “then”. In the proof of Proposition 1.3.4, “Proposition 8.4.7” should be “Proposition 8.4.7 from Analysis I”.
- Page 15: In Definition 1.4.1, “increasing” should be “strictly increasing”.
- Page 19: In Case 1, “Proposition 8.4.7” should be “Proposition 8.4.7 from Analysis I”.
- Page 20: In the analysis of Case 1 of the proof of Theorem 1.5.8,
should be
(two occurrences).
- Page 21: In Exercise 1.5.2, “Lemma 8.4.5” should be “Lemma 8.4.5 from Analysis I”. In Exercise 1.5.3, “Theorem 9.1.24” should be “Theorem 9.1.24 from Analysis I”.
- Page 25: In Lemma 2.2.1, the hypothesis that
is a metric space should be included.
- Page 26: 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 29: 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 36: 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 42: In Exercise 3.1.5, add the hypothesis
.
- Page 45: In Exercise 3.2.2(c), Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I.
- Page 47-48: 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 57: In Exercise 3.7.2,
should be
(because we do not assume
to be bounded).
- Page 66: In Theorem 4.1.6(e), the interval
should be assumed to be non-empty.
- Page 68: In Definition 4.2.4, it is slightly better to use
rather than
(even though they can both be shown a posteriori to be well defined and equal to each other).
- Page 71: In Exercise 4.2.7, “let
be real numbers” should be “
be a real number”, and “Lemma 7.3.3” should be “Lemma 7.3.3 from Analysis I”.
- Page 72 onwards: any appearance of colons in limits, such as
, should be replaced with semicolons for consistency.
- Page 85: At the end of the paragraph following Definition 4.6.12, after “in the usual manner by the formula
“, add: “In the language of abstract algebra, this fact (together with Lemma 4.6.6) tells us that the complex numbers form a field, much like the rational numbers and real numbers.”
- Page 86: In the final display in Lemma 4.6.14, the parentheses should be matched in size.
- Page 88?: In the proof of Theorem 4.4.1, in the discussion of absolute convergence, replace the double series
with the single series
. Similarly, when invoking Fubini’s theorem for series, refer to the absolute convergence of the single series rather than the double series.
- Page 89: In the proof of Lemma 4.7.3, “
increasing” should be “
is strictly increasing”.
- Page 92: In Exercise 4.7.10(c), “Exercise 5.4.3” should be “Exercise 5.4.3 from Analysis I”
- Page 93: replace “not real analytic” with “not real analytic everywhere”, and “do not have power series” with “do not have power series around every point”.
- Page 95: replace
with
.
- Page 98: In Exercise 5.2.3,
should be
.
- Page 99: Expand Exercise 5.2.5 to “Find a sequence of continuous periodic functions
which converge in
to a discontinuous periodic function
in the sense that
as
(extending the
(semi-)norm to discontinuous periodic functions in the obvious fashion), but which do not converge in
to any continuous periodic function.” (and retain the same hint).
- Page 100: In Lemma 5.3.5,
should be
.
- Page 103: After the proof of Lemma 5.4.4: “Lemma 5.4.4(iii)” should be “Lemma 5.4.4(c)”. During this proof: in the long display, the final
should be
. Similarly, on the next page in the proof of Lemma 5.4.6,
should be
.
- Page 104: 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 93: “Napoleons” should be “Napoleon’s”. (This has already been corrected in some versions.)
- Page 108: In the proof of Theorem 5.5.4,
should be
(two occurrences).
- Page 110: In Definition 6.1.10, add that the
are real numbers, and add that to be completely formal, one could view a matrix
as a function
from the set
to
, though we will not use this notation in the text.
- Page 119: In Exercise 6.1.4, give the definition of the norm
(repeating the definition given in Definition 6.2.2).
- Page 120: In Lemma 6.2.1,
should be required to be an element of
. In example 6.2.3, all colons in the limits should be semicolons.
- Page 122: At the end of Section 6.2, remark that it is common to abbreviate
as
when there is no chance of confusion. Add an exercise 6.2.3: “Let
be a subset of
, let
be a function, let
be an interior point of
and let
be the components of
. Show that
is differentiable at
if and only if all of the
are differentiable at
.
- Page 122: In the discussion after Lemma 6.2.4, replace
with
, and remark instead that the derivative
of a function on
is not uniquely defined (only the action of
on the x-axis is unique).
- Page 128: In the hint for Exercise 6.3.1, replace Exercise 6.2.1 with Exercise 6.2.2.
- Page 132: In the proof of Theorem 6.5.4, “the
variable” should be “the
variable”.
- Page 135-136: In Exercise 6.6.1, Corollary 10.2.9 should be “Corollary 10.2.9 from Analysis I”. In Exercise 6.6.7, Lemma 7.3.3 should be Lemma 7.3.3 from Analysis I. In the second paragraph of Section 6.7, “
is not invertible” should be “
is not invertible”.
- Page 147: Before Theorem 7.1.1, in the parenthetical, add the clarification “, and will adopt the convention that
for all
.”
- Page 147: in the discussion of the boolean algebra property, add the remark that we adopt the convention that the empty intersection (i.e., the intersection with
the empty set) is the whole space
- Page 148: After Definition 7.2.1, “open intervals” should be “open bounded intervals”.
- Page 149: In the first sentence after Definition 7.2.4,
should be
. In Lemma 7.2.5(vi), the hypothesis of measurability should be dropped. After Definition 7.2.3, “for every box
” should be “for every box
“.
- Page 150: In Propsition 7.2.6, the hypotheses $b_i \geq a_i$ should be added.
- Page 153: The final claim of Corollary 7.2.7 is more logically placed as the final claim of Proposition 7.2.6.
- Page 154: In the proof of Corollary 7.2.7, “Corollary 6.4.14” should be “Corollary 6.4.14 from Analysis I”; in Example 7.2.9, “Corollary 8.3.4” should be “Corollary 8.3.4 from Analysis I”. In Example 7.2.11, “
has outer measure” should be “
has outer measure”.
- Page 155: In Exercise 7.2.3(a), note that the notion of limit in the extended real number system is defined in Exercise 2.5.5. In Exercise 7.2.4,
should be
, and “covering
by some translates of
” should be “packing
by disjoint translates of
“. In Exercise 7.2.1,
should be
, and
should be
.
- Page 156: In the paragraph before the first display (after Proposition 7.3.1), “Section 8.4” should be “Section 8.4 from Analysis I”.
- Page 160: in the proof of Lemma 7.4.8, just before “we see from Lemma 7.4.5”,
and
should be
and
respectively.
- Page 161: “… on our wish list is (a)” should be “… on our wish list is (i)”.
- Page 162: In the proof of Lemma 7.4.10, first paragraph, “Corollaries 8.1.14, 8.1.15” should be “Corollaries 8.1.14, 8.1.15 from Analysis I”. In the proof of Lemma 7.4.11, the two occurrences of “countable” be “countable or finite”.
- Page 164: In Lemma 7.5.3, specify that the box
lies in
. In Corollary 7.5.7, “then so is” should be “then so are”.
- Page 166: In the proof of Lemma 7.5.10, Definition 6.4.6 should be “Definition 6.4.6 from Analysis I”.
- Page 167: In the first sentence of Chapter 8, “In Chap. 11” should be “In Chap. 11 of Analysis I”. In the final sentence of Example 8.1.2, add the hypothesis that
is non-empty. Then add “Of course, if
is empty, then there is only one function from
to
– the empty function – and it will also be simple since its image is empty.”
- Page 167-168: In Example 8.1.7, all integrals should be over
rather than
.
- Page 169: In Example 8.1.7, “simple integral” should be “Lebesgue integral”.
- Page 170: In the proof of Proposition 8.1.10, after the second and third displays, the two occurrences of “disjoint subsets” should be “disjoint measurable subsets”.
- Page 172: In Remark 8.2.3, “Definition 11.3.2” should be “Definition 11.3.2 from Analysis I”.
- Page 173: In Remark 8.2.7, “the collective set of points” should be “collective sets of points of positive measure”.
- Page 176: In the proof of Lemma 8.2.10, after the third display, “Proposition 8.1.10(db)” should be “Proposition 8.1.10(b)”, and “Proposition 8.1.9(d)” should be “Proposition 8.1.10(d)”. In the statement of Lemma 8.2.13, “non-negative functions” should be specified as “non-negative measurable functions”.
- Page 177: In the proof of Lemma 8.2.13, right after the first display, add a parenthetical remark: “noting that
is increasing with respect to
” . In Lemma 8.2.15, the subscript
for
should be changed to another symbol, e.g.,
.
- Page 178: In the hint of Exercise 8.2.6, “Corollary 11.6.5” should be “Corollary 11.6.5 from Analysis I”. In Exercise 8.2.8, the set
can be taken in fact to be of measure zero. (Then one should add at the end of the hint “now send
to zero”.)
- Page 179: In Exercise 8.2.9, “for all
” should be “for some
” in the set builder notation.
- Page 181: In the first and third display of the proof of Theorem 8.3.4, parentheses can be enclosed around
,
,
, and
for clarity. Similarly for the penultimate display in the proof of Lemma 8.3.6 on page 183.
- Page 182: 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
. (This latter error has already been corrected in some versions of the text.)
- Page 183: In Exercise 8.3.3,
should be
.
- Page 186: In Theorem 8.5.1, “Then there exists” should be “Then there exist”.
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. This has unfortunately caused some ambiguity in the Analysis II text when an Analysis I result was referenced; the above errata attempt to clarify such cases, but the list may be incomplete.
Thanks to Rona Alempour, William Ballard, Quentin Batista, Biswaranjan Behera, José Antonio Lara Benítez, Dingjun Bian, Petrus Bianchi, Philip Blagoveschensky, Carlos, cebismellim, William Clark, William Deng, Jonas Esser, EO, Florian, Aditya Ghosh, Gökhan Güçlü, Yaver Gulusoy, Kyle Hambrook, Minyoung Jeong, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Brett Lane, Matthis Lehmkühler, Yingyuan Li, Zijun Liu, Rami Luisto, lyyryan, Jason M., Matthew, Manoranjan Majji, Geoff Mess, Guillaume Olikier, Jorge Peña-Vélez, Cristina Pereyra, Olli Pottonen, Issa Rice, Linda Riewe, Frédéric Santos, SkysubO, Jorge Silva, 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.
504 comments
Comments feed for this article
7 February, 2024 at 1:45 am
Anonymous
Dear Professor Tao,
On page 54 it is mentioned that “we can rearrange limits and integrals (on compact intervals
)”, but I don’t see why the same proof wouldn’t work if we replace
by any bounded interval
.
[Yes, the argument extends to this setting also, with appropriate notational changes. -T]
20 February, 2024 at 6:45 am
Yingyuan Li
Dear Professor Tao,
I am a bit confused about the definition of diophantine number in Exercise 8.2.7 in page 178. It says “Call a real number
diophantine if there EXIST real numbers
such that
for all nonzero integers
and all integers
.”
But I have notice that there is another definition on another webpage in this blog:
it says “Exercise 3. A real number
is Diophantine if FOR EVERY
there exists
such that
for every rational number
. ”
The main difference is the quantifier of the exponents: the former being
and the latter being
.
So I am wondering which one is correct because this would make the proof rather different.
In addition, the hint in the book says “one can take
and
to be rational”. However, it seems that it is only helpful to consider the set of diophantine numbers as a countable union of some sets. But we are actually proving that the set of non-diophantine numbers has measure zero. And I can not see how this point of the hint is helpful to the proof. It seems that the proof can be done at all without this point of hint if we adopt the definition in the book.
Thank you very much!
Best wishes,
Yingyuan Li
[There is more than one notion of a Diophantine number; the precise specification depends on the particular application, though they all have broadly the same meaning of “not well approximated by a rational number”. The restriction to rationality helps establish that the set of diophantine or non-diophantine sets is measurable, although strictly speaking one does not need to first establish measurability to check that a set is null if one can contain it inside sets of arbitrarily small measure. -T]
22 February, 2024 at 5:50 am
Yingyuan Li
Dear Professor Tao,
Here might be a possible erratum.
In page 179 (Analysis II, 4th edition, Springer), Exercise 8.2.9, in the hint, though there has been an erratum to the 3rd edition:
“the “for all
” should be moved inside the set builder notation
, thus using
instead.”
I think it might still be incorrect in the 4th edition.
What we are proving might actually be that: for any positive integer
, we can find an
such that
![m\left( \bigcup_{n=N_m}^{\infty}\left\{ x\in[0,1]:f_n(x)>1/m \right\} \right) \leq \varepsilon/2^m \\](https://cdn.statically.io/img/s0.wp.com/latex.php?latex=m%5Cleft%28+%5Cbigcup_%7Bn%3DN_m%7D%5E%7B%5Cinfty%7D%5Cleft%5C%7B+x%5Cin%5B0%2C1%5D%3Af_n%28x%29%3E1%2Fm+%5Cright%5C%7D+%5Cright%29+%5Cleq+%5Cvarepsilon%2F2%5Em+%5C%5C+&bg=ffffff&fg=545454&s=0&c=20201002)
(so as to further show that the set
is the set
as desired).
Hence when we translate the union notation into quantifier inside the set builder, we should use “for some” rather than “for all”.
So I think it should be
instead.
Best wishes,
Yingyuan Li
[Erratum added – T.]
22 February, 2024 at 6:18 am
Yingyuan Li
Or simply using
instead.
28 February, 2024 at 7:29 am
Yingyuan Li
Dear Professor Tao,
Here are some possible minor errata.
In page 181 (Analysis II, 4th edition, Springer), in the proof of Theorem 8.3.4, in the first and third display, parentheses should be added to the integrals so that
should be
,
be
.
Similarly in page 183, in the proof of Lemma 8.3.6, in the last but one display,
should be
.
By the way, I am glad that I have finished reading the Analysis I and II series today. Thank you very much again for your warm engagement with readers.
Best wishes,
Yingyuan Li
[Suggestion added, thanks – T.]
2 March, 2024 at 9:13 pm
Anonymous
Dear Professor Tao,
In Theorem 1.5.10, in case 1 where
, it seems that the proof doesn’t utilize the condition that there is no finite subcover. Does this mean that
is impossible regardless of whether or not there is a finite subcover? [Yes – T.]
Furthermore, in case 2 where
, the proof builds a sequence using a recursive procedure. I believe this involves the use of the axiom of choice. Am I correct? [Yes, though one can use weaker forms of choice, such as countable choice, here. -T]
Theorem 1.5.10 aims to prove that every open cover of a compact set has a finite subcover.
9 March, 2024 at 10:42 am
laamoum
Dear Professor Tap
Thr definition 6.1.10 on page 130 is not mathematically correct because it uses the notion of “objects” which is not defined in mathematics.
The correct definition of a matrix A of order nxm with coefficient a_i, j in a field IK is an application of [[1, n]] x[[1, m]] to IK which has (i, j) associates a_i, j.
This automatically gives operations on the matrix and vector space structure of their set
[Note added in the erratum – T.]
14 March, 2024 at 7:13 am
Anonymous
Dear Professor Tao,
I have two questions I want to ask you:
With the definition of compact topological spaces, i think that if
is compact then every
is also compact. Is that right?
Hope to receive answer from you.
14 March, 2024 at 8:27 pm
Anonymous
In context as general as topological spaces, one does not have the notion of metrics. So for these situations, sequential compactness needs general definition.
But the most general one, one still can utilize is the defn. (of compactness) by the existence of finite sub-cover.
T. N.
14 March, 2024 at 8:46 pm
Anonymous
i am confused with excercise 2.5.12 and 2.5.13. What are statements to prove?
4 April, 2024 at 11:05 pm
Olli Pottonen
Dear Professor Tao,
In Exercise 4.2.7 (page 82) it is unclear whether negative x is allowed: on one hand 0 < x < r, on the other x in (-r, r). I believe the first inequality should be just 0 < r.
Also “Lemma 7.3.3” should be “Lemma 7.3.3 of Analysis I”.
[Errata added, thanks – T.]
23 April, 2024 at 7:32 am
Anonymous
Lemma 7.5.3 shouldn’t open box in the lemma exclude volume 0 box?
every open set can be composed with countable open boxes.
which can be used to prove preimage of every open box is measurable then preimage of every open set is measurable.
but I can’t prove the other direction when open box can be a zero volume and not an open set.
23 April, 2024 at 10:39 am
Terence Tao
The only open box of zero volume is the empty set, which is both open and measurable (and the preimage of the empty set is again the empty set).
25 April, 2024 at 12:08 am
Anonymous
Oh. I mistakenly thought, for example x axis (which is not empty) in 2d plane is an open box, from the explanation after the definition 7.2.1 (When one of an interval is empty (a_i = b_i), it is still considered an open box). But upon reading your answer, i found that the text clearly states it’s an empty box.
Thank you!
29 May, 2024 at 5:39 am
Yingyuan Li
Dear Professor Tao,
Here are some minor errata in Analysis II, 4th edition, Springer.
Page 12, in Exercise 1.2.2, the “Lemma 8.4.5”;
Page 14, in the proof of Proposition 1.3.4, first paragraph, the “Proposition 8.4.7”;
Page 19, in Case 1, the “Proposition 8.4.7”;
Page 21, in Exercise 1.5.2, the “Lemma 8.4.5”; and in Exercise 1.5.3, the “Theorem 9.1.24”;
Page 156, in the paragraph before the first display, the “Sect. 8.4”;
All of the above should be added “from Analysis I“.
Best wishes,
Yingyuan Li
[Corrections added, thanks – T.]
8 July, 2024 at 4:43 am
William Clark
This is a suggestion for the third edition in the proof of Theorem 4.4.1 on page 88.
It is written: “the series
is convergent, which means that the sum
is absolutely convergent.” This is true, but lead me to believe something which I believe is false:
is absolutely convergent allows us to apply Fubini.
In my opinion, the sum
being absolutely convergent means that
is convergent, which does not imply that
is convergent. In particular, I believe this illustrates the same mistake that is illustrated in example 1.2.5 of Analysis 1.
In my opinion, the series
being convergent is what is needed to be able to apply Fubini’s theorem. On the other hand, the sum
is not sufficient for applying Fubini’s theorem, as Example 1.2.5 shows.
[Erratum added, thanks – T.]