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An epsilon of room: pages from year blog

Terence TaoVolume 1 (Real analysis): Graduate Studies in Mathematics 117, American Mathematical Society, 2010

ISBN-10: 0-8218-5278-7

ISBN-13: 978-0-8218-5278-1

Volume 2: American Mathematical Society, 2011ISBN-10: 0-8218-5280-9

ISBN-13: 978-0-8218-5280-4

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness” and “Poincaré’s legacies“.

Online versions of both volumes can be found here and here respectively.

Vol I. was reviewed for the Bulletin of the London Mathematical Society by Jonathan Partington here.

Errata for Vol. I:

- In some early printed versions, the section numbering of references in the preface are either incorrect or missing. (This only affects a very small number of copies.)
- Page 6: In Exercise 1.1.2, “unique” should be “maximal”. (It should also be stated that in this text, manifolds are assumed to be second countable.)
- Page 7: In Exercise 1.1.8, add “and let the extension be the measure constructed in the proof of that theorem” to the end of the first sentence. The portion of the exercise regarding how the assumption of finite measure on may be dropped in the sigma-finite case is incorrect and should be deleted.
- Page 15: In (1.5), should be .
- Page 20: In the statement of Theorem 1.2.4, replace by “ is measurable”. Then add “If is finite, then and is finite”. Also replace “ are finite”with “” are finite”. In Corollary 1.2.5, needs to be finite rather than -finite, and in Corollary 1.2.5 (iii), replace “” with ““. After Corollary 1.2.5, add “When is -finite rather than finite, the three claims (i), (ii), (iii) in Corollary 1.2.5 are no longer equivalent; however, a modification of the above arguments shows that claim (ii) holds if and only if for some measurable with at least one of absolutely integrable. We take this to be the definition of absolute continuity in the -finite case.”
- Page 21: In Corollary 1.2.5, should be .
- Page 33: In (1.18), should be .
- Page 34: In Exercise 1.3.2 (iv), in the case the additional hypothesis that are non-negative is required.
- Page 39: In Exercise 1.3.10, and should be and respectively.
- Page 41: In the proof of Theorem 1.3.16, “E” should be in math mode (second paragraph) and “g” should be in math mode (end of third paragraph). Just before (1.28), should be .
- Page 42: In the second to last line, should be .
- Page 50: In Exercise 1.4.6, “this inequality” should be “this equality”, and “non-empty finite measure” should be “non-zero finite measure”.
- Page 53: In Remark 1.4.15, “next set of notes” should be “next section”.
- Page 55: In Exercise 1.4.18(iv), “take adjoints of (ii)” should be “take adjoints of (iii)”.
- Page 60: When the notion of isomorphism of normed vector spaces is defined shortly before Exercise 1.5.1, it should be remarked that this is a looser notion than the isometric notion of isomorphism for Hilbert spaces employed after Exercise 1.4.1. (Different categories of isomorphism.) Also, in the definition of equivalence, “continuous linear transformation” should be “bicontinuous linear transformation”. In Remark 1.5.5, one of the “H”s is incorrectly not set in math mode.
- Page 64: In the second display in the proof of Proposition 1.5.7, should be .
- Page 65: In the proof of the complex case of the Hahn-Banach theorem, should take values in rather than .
- Page 65: In Exercise 1.5.14, should be . Two lines after 1.5.7, “we see that has norm at most 1″ should be “we see that has norm at most 1″. After Exercise 1.5.14, the double dual should be described as the space of <I>continuous</I> linear functions on .
- Page 66: In Exercise 1.5.15(i), the colon after the first appearance of should be deleted.
- Page 67: In the proof of Theorem 1.5.13, should be .
- Page 68: In Remark 1.5.15, add “, with is now required to be non-zero”. In Exercise 1.5.18, X should be a real vector space rather than a complex one, and “usual Hahn-Banach theorem” should be “usual Hahn-Banach theorem for real normed vector spaces”.
- Page 74: In the proof of (ii) implying (i) in Theorem 1.6.8, a closed parenthesis should be added after “converging to “.
- Page 81: In Definition 1.6.29, “partially ordered set” should be “pre-ordered set”.
- Page 89: In the third part of Exercise 1.7.4, some reasonable notion of “operator norm” for a nonlinear operator in order to make the question well-posed. One choice that works is the Lipschitz norm, but the reader is invited to experiment with other choices as well. In the proof of Theorem 1.7.5, “must be dense in a ball” should be “one of the must be dense in a ball”.
- Page 90: In Corollary 1.7.7, Y should be assumed to be Banach space and not merely a normed vector space.
- Page 92: In the proof of Theorem 1.7.12, should be , and similarly should be , and should be , should be , and should be .
- Page 93: In Exercise 1.7.6, the Hahn-Banach theorem is actually not necessary for this exercise. The conclusion “ is an isomorphism” should instead read “ is a linear homeomorphism”.
- Page 94: In Exercise 1.7.7, “we can ensure that” may be clarified to “the above statement remains true if we impose the additional condition that”.
- Page 95: In Example 1.7.18, both occurrences of should be . In Remark 1.7.17, “are uncomplemented” should be “is uncomplemented”, and “the appendix” should be “Section 1.7.4”.
- Page 98: In the proof of Theorem 1.7.22, all occurrences of should be replaced by .
- Page 100: “Phi, Isett” should be “Phil Isett”.
- Page 103: Before Exercise 1.8.4, in the definition of an ultrafilter, should be .
- Page 104: In Exercise 1.8.8, “basis” should be “base”.
- Page 111: In Exercise 1.8.21, in the optional fifth part, the hypothesis “Let be a first-countable topological space” is missing.
- Page 112: In the last bullet point of Definition 1.8.18, should be .
- Page 113: In Example 1.8.21, “” should be “ and “.
- Page 119: At the end of Exercise 1.9.3, add the sentence “Such spaces are known as
*locally convex topological vector spaces*.”. In Example 1.9.4, “topological vector space” should instead read “topological space, but not a topological vector space (because multiplication is not continuous)”, and the final sentence of the example should be deleted. - Page 120: In Exercise 1.9.7, one needs to require the additional hypothesis that is finite. Also, the parenthesis after the epsilon in the definition of needs to be moved to the left of the sign.
- Page 123: In the second and third parts of Exercise 1.9.13, V (and hence and ) need to be assumed to be normed vector spaces.
- Page 124: In Remark 1.9.15, should be .
- Page 125: In Remark 1.9.17, “proper chain” should be “proper well-ordered chain” throughout.
- Page 125: In Exercise 1.9.19, the third item is
**incorrect**and should be deleted. - Page 135: should be . “the empty set has sup 0” should be “the empty set has sup 1”. In Exercise 1.10.3, “the rationals” should be “the set consisting of the rationals “. To put it another way, is the coarsest topology such that every set that is open in , is open in , and such that is also open.
- Page 136: In Exercise 1.10.4, “on this finite set” should be “on this countable set”. In Definition 1.10.2, should be .
- Page 137: In Proposition 1.10.4, the hypothesis that X is sigma-compact may be deleted, by removing all references to the compact set K in the proof (and also deleting the last sentence of the proof). In Exercise 1.10.7, add the following parenthetical remark: “(This question is easier to prove if one assumes that every non-empty open set has positive measure, but it is also possible to solve the question without this additional hypothesis, by working in the “support” of the measure, that is to say a closed set in which every non-empty open set has finite measure, and then using the Tietze extension theorem.)”. This exercise should thus also be moved to after the Tietze extension theorem.
- Page 140: In Exercise 1.10.10, “Then there exists” should be “Show that there exists”. In the proof of Theorem 1.10.8, Theorem 1.10.5 should be Exercise 1.10.10.
- Page 141: Add the following exercise: “Let be a locally compact Hausdorff space, and let be an open cover of . Show that there exist compactly supported continuous functions supported on for each with for all (with only finitely many of the terms on the left-hand side non-zero for each ).”
- Page 146: After “the class of measurable sets is a Boolean algebra”, add “and that is finitely additive on this Boolean algebra”. The sentence fragment “Each in this supremum is supported in some closed subset of ” should be replaced by “For each , each in this supremum is bounded by plus a continuous function between and for some closed subset of “.
- Page 148: In Lemma 1.10.15, “functions” should read “functionals”, and “” should read ““. Exercise 1.10.15 should read as follows: “Show that among all possible choices for the functionals appearing in the above lemma, there is a unique choice which is
*minimal*in the sense that for any other functionals obeying the conclusions of the lemma, and for all non-negative .” - Page 149: After the definition of vague convergence, a remark should be added that an application of the uniform boundedness principle (and Exercise 1.10.7) shows that vague convergence of to is equivalent to the being uniformly bounded in <b>and</b> that for all , however the uniform boundedness aspect cannot be dropped (consider for instance the sequence on the real line).
- Page 152: In Exercise 1.10.25, should be . In Exercise 1.10.24, “Then” should be “Show that”.
- Page 154: In Exercise 1.10.30, the phrase “on compact subsets of ” is redundant and can be deleted.
- Page 155: In Exercise 1.10.35, insert “when is self-adjoint” after the second display.
- Page 161: In (1.82), should be . In Remark 1.11.4, should be . Also, strictly speaking, one should dispose of the degenerate case when or , though this case is easy since non-trivial holomorphic functions cannot vanish on a line.
- Page 162: In Exercise 1.11.4, and should be and respectively, and similarly for .
- Page 164: In the fourth proof, “analytic function of ” should be “analytic function of “. In Exercise 1.11.5, add “up to almost everywhere equivalence”.
- Page 166: In the fourth display, the final should be . In Example 1.11.6, the first should be . In Exercise 1.11.7, the dyadic Lorentz norm defined in the exercise is not quite a quasi-norm, because the homogeneity axiom is not satisfied. Instead, weakened version of the quasi-norm axioms in which the homogeneity axiom is replaced by a quasi-homogeneity axiom for some independent of . homogeneity rather than quasihomogeneity by working with the non-dyadic Lorentz norm instead of the dyadic Lorentz norm, which is equivalent up to constants with the dyadic Lorentz norm, although this was not the intent of the exercise.)
- Page 167: In Exercise 1.11.8, should be . In the last , the condition needs to be imposed.
- Page 168: In Exercise 1.11.10, should take values in rather than , and should be replaced by .
- Page 169: In the first sentence of Section 1.11.3, “these notes” should be “this section”. In Exercise 1.11.13, add that is the dual exponent defined by , and is allowed to be negative in this exercise. Also, it is understood that an assertion such as is false if is not absolutely integrable.
- Page 170: In Remark 1.11.8, the second semicolon should be a comma.
- Page 171: “it is in fact convex in all of ” should read “it is also convex in the triangular region “. In Exercise 1.1.16, “ (resp. )” should be “ (resp. )”, and there is an extra whitespace after “a.e.”.
- Page 172: In the definition of a sublinear operator, the additional condition should be added in addition to (with a similar modification to Remark 1.11.11). “ is a family of linear operators” should be “ is a family of sub-linear operators”, and should map to -valued or -valued” functions rather than just -valued functions. In Exercise 1.11.17, the requirement that p be finite or X has finite measure should be imposed through the entire exercise, not just for the uniqueness aspect. As such, when defining strong and weak type, one should only use the second bullet point rather than the first or third bullet point (unless one has finiteness of p or of the measure of X).
- Page 173: Just before Theorem 1.11.10, Marcinkeiwicz should be Marcinkiewicz. The sentence “We say that a linear operator is of strong type …” is redundant and may be deleted.
- Page 174-175: All occurrences of “(1.91), (1.94)” should be “(1.93), (1.94)”.
- Page 176: In the first display, should be . In Remark 1.11.12, there should not be a C in the subscript of (two occurrences). In Exercise 1.11.18 and Exercise 1.11.19, one also needs to add the hypothesis .
- Page 177: In Exercise 1.11.21, should be .
- Page 180: In Exercise 1.11.24, should be .
- Page ???: In Exercise 1.11.26,the hypothesis should be .
- Chapter 12: In general, the discussion in this chapter should be restricted to sigma-compact LCA groups (due to the reliance on Fubini’s theorem and the Riesz representation theorem, both of which become quite delicate outside of this setting.)
- Page 187: In Exercise 1.12.3, should be .
- Page 188: In Exercise 1.12.7(b), the question is technically solvable as stated, but the “Conversely” portion of the question has a trivial answer as currently written. It should read “For every , , and neighbourhood of the identity, there exists supported on such that ” (i.e. the requirement that has small support is missing). [Actually, this part of the exercise is rather tricky and is not strictly needed for the rest of the exercise; I will probably split it off into a separate exercise in the next edition of this text.]
- Page 190: In Exercise 1.12.10, the requirement that is non-trivial is redundant and may be deleted. For clarity “almost every ” should be “almost every “. In Exercise 1.12.9(g), “Note that this identification is not unique” should be “We caution that in general, this identification is not unique”.
- Page 194: In the “Unitarity” component of Corollary 1.12.5, “Thus the” should simply be “The”.
- Page 196: In Exercise 1.12.20, should be . In Exercise 1.12.21, should be .
- Page 198: In the display in Exercise 1.12.25, should be . The final word “that” on this page should be deleted.
- Page 201: In the final display of Exercise 1.12.36, and should be and respectively. Also, in (1.109), and should be and respectively.
- Page 214. The topology placed on given in the paragraph before Exercise 1.13.2 is not suitable for the purposes of this section (it is not locally convex, or even a topological vector space). To fix this, replace the sentences starting with “Because of this…” with “Because of this, we are able to give a (very strong) topology as follows. Call a seminorm on
*good*if it is a continuous function on for each compact (or equivalently, the ball is open in for each compact ). We then give the topology defined by all good seminorms. Clearly, this makes a (locally convex) topological vector space.” - Page 215: Exercise 1.13.3(iii) is
**incorrect**and should be replaced with the following: “(iii) As an additional challenge, construct a set such that is an adherent point of , but is not as the limit of any sequence in .” Exercise 1.13.4(iii) should then be replaced with “Show that a linear map from the space of test functions into a topological vector space generated by some family of seminorms (i.e., a*locally convex topological vector space*) is continuous if and only if it is sequentially continuous (i.e. whenever converges to in , converges to in ), and if and only if is continuous for each compact . Thus while first countability fails for , we have a serviceable substitute for this property.”. Finally, Exercise 1.13.4 (viii) is trivial and should be deleted. - Page 215: In Exercise 1.13.4(v), the map T needs to be assumed to be continuous.
- Page 216: The definition of approximation to to the identity before Exercise 1.13.5 needs to be strengthened, in particular “converge uniformly to zero away from the origin, thus for all ” should be replaced by “has supports shrinking to the identity, thus for each , is supported on for sufficiently large “.
- Page 217: In the second example after Exercise 1.13.7, “Note that this example generalises the previous one” should be “Note that this example generalises the previous one (in the unsigned or absolutely integrable cases, at least)”.
- Page 224: In the third bullet point of Exercise 1.13.25, “some compactly supported distributions” should be “some compactly supported distributions “.
- Page 225: In the final bullet point of Exercise 1.13.26, “show that ” should be “show that “, and should also be preceded by a comma.
- Page 230: In the final display, should be .
- Page 232: In Exercise 1.13.37, all the fundamental solutions are missing a minus sign (this is ultimately due to the refusal to put a minus sign in the definition of the Laplacian, as alluded to on page 229). In Exercises 1.13.38-1.13.40, the Laplacian should be understood to be with respect to the spatial variable (i.e. it is not the spacetime Laplacian). In Exercise 1.13.38, the factor of in the definition of should just be .
- Page 233: In Exercise 1.13.39, the factor of in the definition of should just be . In Exercise 1.13.40, “wave equation ” should be “wave equation ″, and “Schwartz functions ” should be “Schwartz functions “.
- Page 238: In the display before Remark 1.14.1, the symbols should be replaced by . Similarly for the definition of L after Exercise 1.14.3. In the definition of , “derivatives of order ” should be “derivatives of order up to “.
- Page 241: In Exercise 1.4.12(2), the additional hypothesis that is missing.
- Page 242: In Exercise 1.4.14, the signs are reversed in the formulae for and for (i.e. there should be a negative sign in the former and a positive sign in the latter, rather than the other way around. Also, in (iii), the formula for has both a reversed sign and a missing term; it should be.
- Page 243: In Exercise 1.14.15, Kondrakov should be Kondrachov.
- Page 244: In Exercise 1.14.16, should be .
- Page 245: In Exercise 1.14.18, “is ” should be “is contained in “.
- Page 246: In Theorem 1.14.7, “encluding” should be “excluding”.
- Page 248: In Exercise 1.14.20, “of the form” should be “which are something like”. In Exercise 1.4.22, the hypothesis that is bounded needs to be added.
- Page 249: Before Exercise 1.14.23, “, which we will do in later notes” should be “; see Exercise 1.15.23”. In the last display of Exercise 1.14.21, the norm should be instead. (The case of this exercise is somewhat degenerate, but the result is still true in this case; however, the reader may wish to exclude this case in order to avoid such degeneracy.) Similarly, in the second to last display of the proof of Lemma 1.14.9, should be .
- Page 254: In the second display of Exercise 1.14.36, should be .
- Page 258: Before (1.125): “chain of maximal ideals” should be “chain of prime ideals”. Bougliand should be Bouligand.
- Page 259: In Section 1.15.1, “k-dimensional subspace” should be “d-dimensional subspace”, and in the second paragraph should be throughout.
- page 260: In the fourth bullet point, should be .
- Page 265: In the first paragraph on this page of Section 1.15.2, “measre” should be “measure”.
- Page 268: In the end of the proof of Lemma 1.15.4, should be , and the inequalities and should be swapped.
- Page 271: In the proof of Lemma 1.15.7, “Huausdorff” should be “Hausdorff”. In the second bullet point, should be . In Exercise 1.15.17, “compact support there” should be “compact support and there”.
- Page 288: In Lemma 2.2.18, replace the parenthetical with “the existence of which follows from the fact that can be well-ordered”. In Exercise 2.2.11, the second part does not need to be in red.
- Page 289: In the proof of Proposition 2.2.19, the summation signs in and just after (2.6) should be union signs. Also add the sentence “By incrementing if necessary, we can take to be one of the .”
- Page 303: In (2.14), should be .
- Page 307: In Remark 2.4.22, “Exercises 2.4.3 and 2.4.4” should be “Examples 2.4.6 and 2.4.7”.
- Page 308: In Exercise 2.4.10, should be .
- Page 311:In Definition 2.5.1, should map from to , and should be .
- Page 319: In the last paragraph of 2.6.1, the term should be .
- Page 334: In the proof of (i) implies (ii) in Theorem 2.8.1, should be , and “and all ” should be omitted.

Errata for Vol. II:

- Page 12: “a random complex number in ” should be “a random complex vector in “.
- Page ???: In Proposition 1.5.6, the dimension should be rather than .
- Page 40: In the statement of Proposition 1.5.7, a factor of 2 should be inserted in the right hand side, and the dimension should be rather than . After that proposition, insert “and dividing into whether is larger than or smaller than , and noting also that the claim is trivial for t small” before “…”.
- Page 41: prefactor should instead be (and conversely, should be ), and the final bound of should instead be .
- Page 43: In Lemma 1.6.1, add “Let P be a finite set of points in R^2” in the first sentence, and replace “in the plane (which may or may not be in L)” with “in L, plus some additional open line segments not containing any points in P”. In section 1.6.2, “carve out O(r^2) cell” should be “carve out O(r^2) cells”. In the paragraph starting with “To fix the latter problem…”, add “Note that almost surely the open line segments added will not contain any points of P.” after the parenthetical sentence.
- Page 54: 30.7% should be 30.1% .
- Page 82, before (1.53): should be .
- Page 90, in the display before (1.61): the final should be .
- Page 112: In Remark 1.15.20, “incompleteness theory” should be “incompleteness theorem”.
- Page 177: After (2.19), “existence of a quadratic residue” should be “existence of a quadratic nonresidue”.
- Page 235: The proof of Lemma 2.as stated, because it is not demonstrated that the embedding of the A’ free group into the A group is injective. The proof can be salvaged by constructing the semidirect product first, and then constructing the isomorphism between that product and the free group. Details can be found at this post.
- Page 238: In the last display, should be added at the end, and similarly for the first display on the next page; in the display after that, should also be appended.
- Page 310: “all group elements ” should be “all group elements “.

Thanks to Marcelo Aguiar, Adam Azzam, Farzin Barakat, Lucas Braune, Yunbai Cao, Nick Cook, Alex Dobner, Kun Dong, Sean Eberhard, Sylvester Eriksson-Bique, Federico, Stephen Ge, Wengyin Gan, Julian Gold, Jordy Greenblatt, Robert Hannah, Matthias Hubner, Joe Hughes, Sune Kristian Jakobsen, Zane Li, Dirk Lorenz, Martin Los, Cao Lu, Freddie Manners, Ian Martin, Ricardo Minares, Alexey Muranov, Sujit Nair, Seungly Oh, John Pearson, Qiang, Yudong Qiu, Rex, James Robinson, Agus Seonjaya, Lior Silberman, August Sonne, Dan Stroock, Jon Susice, Yotas Trejos, Anthony Verbitsky, Ben Wallis, Joshua Wilson, wilsonofgordon, xuhmath and Yaoliang for corrections.

## 27 comments

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8 February, 2010 at 2:28 pm

An epsilon of room: pages from year three of a mathematical blog « What’s new[…] Tao I have just finished the first draft of my blog book for 2009, under the title of “An epsilon of room: pages from year three of a mathematical blog“. It largely follows the format of my previous two blog books, “Structure and […]

17 February, 2010 at 12:55 am

גיקדום 17.02.2010 « ניימן 3.0[…] כבר שלוש שנים, כשכל שנה הוא מייצר ספר מהחומר בבלוג. אז השבוע יצא הספר השלישי, שמדבר על אנליזה מודרנית ברמת הסטודנט, בעיות ברמות של […]

28 September, 2010 at 8:58 am

UlrichTerence, any news about the publishing date of this book?

Ulrich

28 September, 2010 at 9:55 am

Terence TaoWe are working on the galley proofs right now. For various reasons the book was delayed by a few months as compared against the previous volume in this series, but hopefully it will still be ready by the end of this year.

21 December, 2010 at 11:21 am

AnonymousIs the second volume expected to contain the notes from your class on Random Matrix Theory (or are they available in pdf somewhere else?)?

21 December, 2010 at 11:28 am

Terence TaoThis will be part of the 2010 book, which I plan to start working in next month.

21 December, 2010 at 11:30 am

Honglang Wangcoungratulations!

21 December, 2010 at 11:42 am

TimothyWill you include the article about group extensions in the 2010 book? E.g. who you talked about pixels and zooming in.

3 March, 2011 at 7:59 am

UlrichTerence, I am trying since weeks to order the book “A epsilon of room 1” but the book store always get “not available yet”, even Amazon is saying this to me. It is already sold out? If yes, will there be an new printing available?

Regards

Ulrich

4 March, 2011 at 7:54 am

Terence TaoI received my own copies of the book about a week ago, so it should be currently available.

23 May, 2012 at 6:48 am

BenI think in theorem 1.2.4 and corollary 1.2.5 (Lebesgue-Radon-Nikodym) you want f:X -> R measurable, not f:X -> C measurable. If f takes complex values then the measure m_f isn’t going to be defined.

[Corrected, thanks – T.]27 May, 2012 at 4:53 am

BenThank YOU for the awesome book!

21 January, 2013 at 4:19 pm

Isaac SolomonOn page 49, Exercise 1.4.2 asks one to show that a Hermitian positive semidefinite matrix gives rise to an inner product. Does this require the Hermitian matrix to be positive definite? Thanks.

21 January, 2013 at 4:50 pm

Terence TaoNo, the exercise also works in the semi-definite case also. (Note that the are not required to be linearly independent.) One way to deal with the semi-definite case is to first construct a degenerate inner product in which some elements have norm zero, and then somehow fix the space to eliminate the degeneracy.

5 February, 2013 at 8:27 pm

Isaac SolomonOn page 65, in the proof of Hahn-Banach (complex case), it says “optimising in \theta, we see that \tilde{\rho} has norm at most 1”. This seems like a typo, and that it should say “we see that \tilde{\lambda} has norm at most 1”.

[Correction added, thanks – T]18 February, 2013 at 8:47 pm

Adam AzzamMinor typo: On Page 103, two lines above Exercise 1.8.4, I believe that “property that for any , exactly one of and lies in ” should instead read “property that for any , exactly one of and lies in ”

[Correction added, thanks – T.]21 February, 2013 at 9:56 pm

Adam AzzamI don’t intend to be pedantic, but on the optional part of Exercise 1.8.21 (Page 111), there is no explicit assumption that the space X is first countable [unlike all the other parts, which explicitly state the relevant topological assumptions]. I know is more or less implied since the problem statement introduces the notion of first countability (and is false otherwise), but I thought you might want to be aware of this.

[Correction added, thanks – T.]4 June, 2013 at 6:39 pm

Yunfeng ZhangDear Prof. Tao,

The hint for the first part of Exercise 1.14.20 (Page 282) does not seem to work. I believe that one correct version of a counterexample could be

with supported on the annulus .

4 June, 2013 at 6:50 pm

Yunfeng ZhangI mean is a nonzero test function which is 1 on and is 0 on .

5 June, 2013 at 8:28 am

Terence TaoFair enough, the hint should be a bit looser, saying that one should try a function that is “something like” though not exactly this particular function. (Your example works; there are other options too.)

6 June, 2013 at 12:22 pm

Adam AzzamMinor typo: In the second display equation of Exercise 1.14.36 on Page 254, I believe that should be .

[Correction added, thanks – T.]7 January, 2014 at 10:54 pm

Adam AzzamOn Page 233, Exercise 1.13.40, I believe what is written “wave equation ” should perhaps be “wave equation ″.

[Correction added, thanks – T.]22 May, 2014 at 6:44 pm

Dimitris NtalampekosOn this webpage, errata item: page 232, “Exercise 1.13.18” should be “Exercise 1.13.38”.

[Corrected, thanks – T.]29 February, 2016 at 2:44 am

Jonathan SetinmannWhat is the reason for studying complex Hilbert spaces in particular, complex $L_p$ spaces ?

I assume that something similar to basic matrix analysis or complex analysis happens here, where the results from the complex case can succesfully be applied to understand the reals case better (specifically, in the first case the fundamental theorem of algebra makes is compelling to analyse a matrix in the complex domain and then descend the results into the real domain and in the second case real integrals can be succesfully evaluated using methods from complex analysis).

These are the only two instances I know of, that motivate doing the work of extending a given theory (real matrix analysis or real analysis in this case) from the real case to the complex case.

In the case of complex Hilbert spaces, I don’t know of any such application, which makes studying the complex case seem a bit like an idle generalization of the real case. What compelling reasons exist, to take the theory of Hilbert spaces ,in particular the theory of the Lebesgue integral for $[0,+\infty)$- resp. $\overline{\mathbb{R}}$-valued functions, from the real case to the complex one ?

29 February, 2016 at 9:37 am

Terence TaoOne major application is quantum mechanics, which is most naturally phrased on a complex Hilbert space: https://en.wikipedia.org/wiki/Dirac%E2%80%93von_Neumann_axioms

As you note, complex Hilbert spaces also have a much better spectral theorem than their real counterparts, which is something one can already see in finite dimensions, when real normal matrices cannot always be diagonalised (e.g. a rotation matrix in two dimensions) whereas a complex normal matrix always can. Basically, with complex methods one can unify the three major linearised dynamics – exponential growth, exponential decay, and oscillation – into a single dynamic of complex exponentiation (with the qualitative behaviour of the dynamics then determined by whether the spectral parameter has positive real part, negative real part, or is purely imaginary).

Related to this, another major advantage of working with complex-valued function spaces is the ability to cleanly use the Fourier transform: https://en.wikipedia.org/wiki/Fourier_transform . More generally, the theory of Fourier integral operators and pseudodifferential operators, as well as the entirety of microlocal analysis, is best phrased for complex-valued functions.

6 March, 2016 at 11:02 am

StudentIt seems that the order how concepts are introduced in the sentences “If $(c_{\alpha})_{\alpha\in A}$ is square-summable, then at most countably many of the $c_{\alpha}$ are non-zero (by Exercise 1.3.4).[…] we can then form the sum $\sum_{\alpha\in A}c_{\alpha}e_{\alpha}$ in an unambiguous manner.”, taken from pp. 55 from the above pdf, is not working. (Or at least I can’t see an immediate way to make them work).

This is because (square-)summability of a sequence, having possibly uncountable terms, is not defined at the point when it is mentioned. This as yet undefined notion of summability then has to be used to show that it can be exchanged with a usual, countable notion of summability (since “most” terms are zero). Also it is not clear, what Exercise 1.3.4, having to do with step functions, has to do with these summability issues.

7 March, 2016 at 12:24 pm

Terence TaoSquare-summability means that the coefficients lie in , which is with counting measure. This was already constructed even when is uncountable.

Note also that Exercise 1.3.4 is distinct from Example 1.3.4.