Last updated: Jun 30, 2022

An epsilon of room: pages from year three of a mathematical blog
Terence Tao

Volume 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, 2011

ISBN-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.. In Exercise 1.1.5, delete the “of” in “and of the algebra of null sets”.
  • Page 7: In Example 1.1.14, \frac{1}{|E|} should be \frac{1}{|X|}.
  • Page 8: In the first paragraph of the proof of Theorem 1.1.17, “see that if” should just be “see that”.
  • Page 9: In Exercise 1.1.8, add “and let the extension \mu: {\cal X} \to [0,+\infty) 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 E may be dropped in the sigma-finite case is incorrect and should be deleted. In Exercise 1.1.10, \mu should be m (two occurrences).
  • Page 15: In (1.5), d\mu should be dm.
  • Page 17: Before (1.9), “c>0 should be c \geq 0“.
  • Page 20:  In the statement of Theorem 1.2.4, replace f \in L^1(X,dm) by “f: X \to {\bf R} is measurable”.  Then add “If \mu is finite, then f \in L^1(X,dm) and \mu_s is finite”.   Also replace “\mu, \nu are finite”with “\mu,m” are finite”.  In Corollary 1.2.5, \mu needs to be finite rather than \sigma-finite, and in Corollary 1.2.5 (iii), replace “\mu(E) < \varepsilon” with “|\mu(E)| < \varepsilon“.  After Corollary 1.2.5, add “When \mu is \sigma-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 \mu = m_f for some measurable f with at least one of f_+,f_- absolutely integrable.  We take this to be the definition of absolute continuity in the \sigma-finite case.”.  Just before Section 1.2.2, add “(One can also complexify this space to obtain a complex vector space of complex finite measures, but we will not use such measures here.)”.   In the proof of Corollary 1.2.5, the reference to Theorem 1.2.2 should instead be to Theorem 1.2.4.
  • Page 22: After Remark 1.2.8, “”singular continuous”” should be “”absolutely continuous””.
  • Page 33: In (1.18), f^r should be |f|^r.
  • Page 34: In Exercise 1.3.2 (iv), in the 0 < p < 1 case the additional hypothesis that f, g are non-negative is required.
  • Page 36: After Exercise 1.3.6, the first sentence should be replaced by “It is easy to see (using the triangle inequality if p \geq 1, and Exercise 1.3.2 otherwise) that the quasi-norm balls B(f,r) := \{ g \in L^p: \|f-g\|_{L^p} < r \} form a base for a topology on L^p.”
  • Page 39: In Exercise 1.3.10, f^p and g^q should be |f|^p and |g|^q 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), \mu_g should be \mu_{\overline{g}}, and add “(in the sense that the real and imaginary parts are absolutely continuous) after “\nu is absolutely continuous”.
  • Page 42: In the second to last line, m=1 should be g=1.
  • 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 51: In Exercise 1.4.8, a right parenthesis is missing after “of \overline{V} and \overline{V}'“.
  • Page 53: In Remark 1.4.15, “next set of notes” should be “next section”.
  • Page 54: In the second and third bullets of Exercise 1.4.16, T^\dagger \mathrm{T} should be T^\dagger T. 
  • Page 55: In Exercise 1.4.18(iv), “take adjoints of (ii)” should be “take adjoints of (iii)”.  In Remark 1.1.14, “Remark 1.26” should be “Remark 1.3.19”.
  • 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. In Exercise 1.5.1, the finite-dimensional vector space should also be understood to be normed.
  • Page 63: In Exercise 1.5.10, “n \times m” should be “m \times n“.
  • Page 64:  In the second display in the proof of Proposition 1.5.7, \lambda(y') - \|y'+v\|_X should be -\lambda(y') - \|y'+v\|_X.
  • Page 65: In the proof of the complex case of the Hahn-Banach theorem, \tilde \lambda should take values in {\mathbf C} rather than {\mathbf R}.
  • Page 65: In Exercise 1.5.14, \|\lambda\|_{X^*}=1 should be \|\lambda\|_{X^*} \leq 1. Two lines after 1.5.7, “we see that \tilde \rho has norm at most 1″ should be “we see that \tilde \lambda has norm at most 1″.  After Exercise 1.5.14, the double dual should be described as the space of continuous linear functions on X^*.
  • Page 66: In Exercise 1.5.15(i), the colon after the first appearance of \overline{Y} should be deleted.  One can also add that \overline{Y} denotes the closure of Y in X.
  • Page 67: In the proof of Theorem 1.5.13, \|x\| should be \|x\|_X. In Exercise 1.5.17, “disjoint sets of positive measure” should be “disjoint sets of positive finite measure”.
  • Page 68: In Remark 1.5.15, add “, with \lambda 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 72: In Definition 1.6.6, after introducing the notion of an isolated point, add “(a point which is open)”.
  • Page 74: In the proof of (ii) implying (i) in Theorem 1.6.8, a closed parenthesis should be added after “converging to x“.
  • Page 79:  In Example 1.6.21, n \to \infty should be n \to +\infty.
  • Page 80: In Example 1.6.22, “sufficiently large x” should be “sufficiently large n“. In Example 1.6.25, “the direct sum f: Z \to X \times Y” should be “the pairing (f,g): Z \to X \times Y defined by (f,g)(z) := (f(z),g(z))“.
  • Page 81: In Definition 1.6.29, “partially ordered set” should be “pre-ordered set”.
  • Page 87: In the last sentence of the proof of the Baire category theorem, B should be B(x_0,r_0).
  • 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 E_n 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.  In (1.64), T_{\alpha_m} should be T_{\alpha_n}.  In the proof of the Corollary, Theorem 1.7.3 should be Theorem 1.7.5.  Remark 1.7.8 is incorrect and should be deleted.
  • Page 92: In the proof of Theorem 1.7.12, nr+n\varepsilon should be n\|f_0\|_Y + nr, and similarly 2nr+2n\varepsilon should be 2n\|f_0\|_Y+2nr, and 2n\|f\|_Y+2n\varepsilon should be 2n (1 + \frac{\|f_0\|_Y}{r}) \|f\|_Y, \frac{5}{2} n should be 2n (1 + \frac{\|f_0\|_Y}{r}), and 5n should be 4n (1 + \frac{\|f_0\|_Y}{r}).
  • Page 93: In Exercise 1.7.6, the Hahn-Banach theorem is actually not necessary for this exercise.  The conclusion “L^* is an isomorphism” should instead read “L^* 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 c_0({\bf N}) should be c_c({\bf N}).  In Remark 1.7.17, “are uncomplemented” should be “is uncomplemented”, and “the appendix” should be “Section 1.7.4”.
  • Page 97: In the first paragraph of Section 1.7.4, add the remark that thanks to Exercise 1.7.9, this provides examples of closed subspaces of Banach spaces that are not complemented.
  • Page 98: In the proof of Theorem 1.7.22, all occurrences of \mu should be replaced by \sigma.
  • Page 100: “Phi, Isett” should be “Phil Isett”.
  • Page 103: Before Exercise 1.8.4, in the definition of an ultrafilter, E \in X should be E \subset X.
  • Page 104: In Exercise 1.8.8, “basis” should be “base”.
  • Page 111: In Exercise 1.8.21, the optional fifth part is incorrect and should be deleted.
  • Page 112: In the last bullet point of Definition 1.8.18, x',x \in x should be x',x \in X.
  • Page 113: In Example 1.8.21, “X=Y={\bf R}” should be “X = {\bf R} \backslash \{0\} and Y = {\bf R}“.
  • 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 \mu is finite.  Also, the parenthesis after the epsilon in the definition of B(f,\varepsilon,r) 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 V^* and (V^*)^*) need to be assumed to be normed vector spaces.
  • Page 124: In Remark 1.9.15, \ell^\infty({\bf N}) should be \ell^\infty({\bf N})^*.
  • 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 129: In Exercise 1.9.25, the first sentence can be deleted (also the word “throughout” should be appended to the last sentence).
  • Page 135:  \hbox{inf} \{ q: x \in K_q \} should be \hbox{inf} \{ q: x \not \in K_q \}.  “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 \{{\bf Q}\} consisting of the rationals {\bf Q}“.  To put it another way, {\mathcal F}' is the coarsest topology such that every set that is open in {\mathcal F}, is open in {\mathcal F}', and such that {\bf Q} 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, {\bf R}^+ should be [0,+\infty).
  • 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: In the second part of Exercise 1.10.11, “normal” should be “normal and Hausdorff”. Add the following exercise: “Let X be a locally compact Hausdorff space, and let (U_\alpha)_{\alpha \in A} be an open cover of X.  Show that there exist compactly supported continuous functions f_\alpha: X \to [0,1] supported on U_\alpha for each \alpha \in A with \sum_{\alpha \in A} f_\alpha(x)=1 for all x \in X (with only finitely many of the terms on the left-hand side non-zero for each x).”
  • Page 146: After “the class of measurable sets is a Boolean algebra”, add “and that \mu_+=\mu_- is finitely additive on this Boolean algebra”.  The sentence fragment “Each f in this supremum is supported in some closed subset K of U” should be replaced by “For each \varepsilon > 0, each f in this supremum is bounded by \varepsilon plus a continuous function between 0 and 1_{K_\varepsilon} for some closed subset K_\varepsilon of U“.
  • Page 148: In Lemma 1.10.15, “functions” should read “functionals”, and “0 \leq I(f) \leq I^+(f)” should read “0, I(f) \leq I^+(f)“.  Exercise 1.10.15 should read as follows: “Show that among all possible choices for the functionals I^+, I^- appearing in the above lemma, there is a unique choice which is minimal in the sense that for any other functionals \tilde I^+, \tilde I^- obeying the conclusions of the lemma, \tilde I^+(f) \geq I^+(f) and \tilde I^-(f) \geq I^-(f) for all non-negative f \in C_c(X \to {\bf R}).”
  • 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 \mu_n to \mu is equivalent to the \mu_n being uniformly bounded in M(X) and that \int_X f\ d\mu_n \to \int_X f\ d\mu for all f \in C_c(X), however the uniform boundedness aspect cannot be dropped (consider for instance the sequence \mu_n =n \delta_n on the real line).
  • Page 152: In Exercise 1.10.25, [{\bf R},{\bf Z}] should be {\bf R}/{\bf Z}.  In Exercise 1.10.24, “Then” should be “Show that”.
  • Page 154: In Exercise 1.10.30, the phrase “on compact subsets of {\bf R}” is redundant and can be deleted.  (In the online version of the text, the claim “converges uniformly” should be corrected to “converges pointwise a.e.”.)
  • Page 155: In Exercise 1.10.35, insert “when x is self-adjoint” after the second display.
  • Page 161: In (1.82), (\pi-\delta) t should be (\pi-\delta)|t|.  In Remark 1.11.4, A,\sigma should be A,\delta.  Also, strictly speaking, one should dispose of the degenerate case when B_0=0 or B_1=0, though this case is easy since non-trivial holomorphic functions cannot vanish on a line.
  • Page 162: In Exercise 1.11.4, f(0+it) and f(1+it) should be |f(0+it)| and |f(1+it)| respectively, and similarly for f(\sigma+it).
  • Page 164: After (1.83), x_m should be x_M (two occurrences).  In the following display, L^p(X) on the left-hand side should be L^p(X^M).  In the fourth proof, “analytic function of f” should be “analytic function of s“.  In Exercise 1.11.5, add “up to almost everywhere equivalence”.  In the display before (1.84), |f| should be |f(x)|.
  • Page 166: In the fourth display, the final \|f\|_{L^{p,\infty}(X)} should be \|g\|_{L^{p,\infty}(X)}.  In Example 1.11.6, the first q>p should be q0 independent of \lambda, f.  homogeneity rather than quasihomogeneity by working with the non-dyadic Lorentz norm (\int_0^\infty (t^n \lambda_f(t)^{1/p})^q \frac{dt}{t})^{1/q} 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, \log(1+|X|) should be \log^{1/p}(1+|X|).  In the last , the condition p_0 \neq p_1 needs to be imposed.
  • Page 168: In Exercise 1.11.10, f^* should take values in [0,+\infty] rather than {\bf R}^+, and t \geq 0 should be replaced by t>0.
  • Page 169: In the first sentence of Section 1.11.3, “these notes” should be “this section”.  In Exercise 1.11.13, add that p' is the dual exponent defined by 1/p +1/p' = 1, and is allowed to be negative in this exercise.  Also, it is understood that an assertion such as |\int_X f 1_{E'}\ d\mu| \leq C \mu(E)^{1/p'} is false if f 1_{E'} 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 [0,+\infty)^2” should read “it is also convex in the triangular region \{ (1/p,1/q) \in [0,+\infty)^2: p \leq q \}“. In Exercise 1.1.16, “Y (resp. X)” should be “X (resp. Y)”, and there is an extra whitespace after “a.e.”.
  • Page 172: In the definition of a sublinear operator, the additional condition |Tf - Tg| \leq |T(f-g)| should be added in addition to |T(f+g)| \leq |Tf| + |Tg| (with a similar modification to Remark 1.11.11).  “(S_\alpha)_{\alpha \in A} is a family of linear operators” should be “(S_\alpha)_{\alpha \in A} is a family of sub-linear operators”, and T should map to {\bf C}-valued or [0,+\infty]-valued” functions rather than just [0,+\infty]-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 S 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, 2^{n\alpha q_\theta - mp_\theta} should be 2^{(n\alpha q_\theta - m p_\theta) x_i}.  In Remark 1.11.12, there should not be a C in the subscript of C_{p_0,p_1,q_0,q_1,\theta,C} (two occurrences).  In Exercise 1.11.18 and Exercise 1.11.19, one also needs to add the hypothesis p_0 \neq p_1.
  • Page 177: In Exercise 1.11.21, kO(1) should be k+O(1).
  • Page ???: In Exercise 1.11.22, add the hypothesis p \leq q.
  • Page 180: In Exercise 1.11.24, x^{r'} should be y^{r'}.
  • Page 181: In Exercise 1.11.26,the hypothesis 1 \leq p \leq \infty should be 1 < p < \infty.
  • 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, dx should be d\mu(x) .  Exercise 1.12.5 is not correct as stated in the non sigma-compact case.  It can be replaced with the following: “Let G be an LCA group with non-trivial Haar measure \mu. Let \tilde L^\infty(G) be the space of equivalence classes of functions f: G \to {\bf C} such that for every set K of finite measure, the restriction of f to K lies in L^\infty(K) with a norm bounded uniformly in K, with two functions in \tilde L^\infty(G) in the same equivalence class if they agree almost everywhere on every set K of finite measure, and with the \tilde L^\infty(G) norm of f equal to the supremum of the L^\infty(K) norms of the restrictions. (For $\sigma$-finite groups G, \tilde L^\infty(G) is identical to L^\infty(G), but the two spaces differ slightly in general.) Show that L^1(G)^* is identifiable with $\tilde L^\infty(G)$.”.
  • 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 f \in C_c(G)^+, \varepsilon > 0, and neighbourhood U of the identity, there exists g \in C_c(G)^+ supported  on U such that \int_G f\ d\mu \geq (f:g) \int_G g\ d\mu - \varepsilon” (i.e. the requirement that g 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 189: In Exercise 1.12.7(g), “range in” should be “takes values in”.
  • Page 190: Exercise 1.12.9(e) is significantly harder than intended, as the proof requires Pontryagin duality (which is stated, but not proven, in this text).  This part of the exercise should therefore be disregarded. In Exercise 1.12.10, the requirement that \mu is non-trivial is redundant and may be deleted.  For clarity “almost every x,y \in G” should be “almost every (x,y) \in G \times G“. 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, \frac{\sin((N+1/2)x)}{\sin(x/2)} should be \frac{\sin(2\pi(N+1/2)x)}{\sin(\pi x)}.  In Exercise 1.12.21, \frac{\sin(nx/2)}{\sin(x/2)} should be \frac{\sin(\pi N x)}{\sin(\pi x)}.
  • Page 197: In Exercise 1.12.24, all occurrences of x_n should be x_d.
  • Page 198: In the display in Exercise 1.12.25, {\bf R}^n should be {\bf R}^d.  The final word “that” on this page should be deleted.
  • Page 200: In Exercise 1.12.33, f(x_j g) should be f * (x_j g).
  • Page 201: In the final display of Exercise 1.12.36, (\xi-\xi_0) and (x-x_0) should be (\xi-\xi_0)^2 and (x-x_0)^2 respectively.  Also, in (1.109), {\mathcal F} D = X {\mathcal F} and {\mathcal F} X = -{\mathcal F} D should be {\mathcal F} D = - X {\mathcal F} and {\mathcal F} X = D {\mathcal F} respectively.
  • Page 203: In Exercise 1.12.37, (1.103) should be (1.111).
  • Page 209: In the display discussing the FFT, the summation should be over x rather than n.
  • Page 214:  The topology placed on C^\infty_c({\bf R}^d) 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 C^\infty_c({\bf R}^d) a (very strong) topology as follows.  Call a seminorm \| \| on C^\infty_c({\bf R}^d) good if it is a continuous function on C^\infty_c(K) for each compact K (or equivalently, the ball \{ f \in C^\infty_c(K): \|f\| < 1 \} is open in C^\infty_c(K) for each compact K).  We then give C^\infty_c({\bf R}^d) the topology defined by all good seminorms.  Clearly, this makes C^\infty_c({\bf R})^d 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 E \subset C^\infty({\bf R}^d) such that 0 is an adherent point of E, but 0 is not as the limit of any sequence in E.”  Exercise 1.13.4(iii) should then be replaced with “Show that a linear map T: C^\infty_c({\bf R}^d) \to X 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 f_n converges to f in C^\infty_c({\bf R}^d), Tf_n converges to Tf in X), and if and only if T: C^\infty_c(K) \to X is continuous for each compact K \subset {\bf R}^d.  Thus while first countability fails for C^\infty_c({\bf R}^d), we have a serviceable substitute for this property.”.  In Exercise 1.13.4 (iv), the constraint 0 < p should be 1 \leq p. Exercise 1.13.4 (viii) is trivial and should be deleted.  In Exercise 1.13.4(v), the map T needs to be assumed to be linear.
  • 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 \sup_{|x| \geq r}|\phi_n(x)| \to 0 for all r>0” should be replaced by “has supports shrinking to the identity, thus for each r>0, \phi_n is supported on B(0,r) for sufficiently large n“.  In Exercise 1.13.5, it would be better to use C_c({\bf R}^d) rather than C_0({\bf R}^d) in the hint.
  • 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 220: After (1.114), \lambda h should be \lambda * h.
  • Page 224: In the third bullet point of Exercise 1.13.25, “some compactly supported distributions” should be “some compactly supported distributions \rho,\lambda“.
  • Page 225: In the final bullet point of Exercise 1.13.26, “show that L” should be “show that \lambda“, and should also be preceded by a comma.
  • Page 227: In Remark 1.13.10, it is the word “both” that should be italicised, rather than “growth”.
  • Page 230: In the final display, t^d should be t^{-d}.
  • Page 232: In Exercise 1.13.37, all the fundamental solutions K 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 x (i.e. it is not the spacetime Laplacian).  In Exercise 1.13.38, the  factor of |x-y|^2 in the definition of K_t should just be |x|^2.
  • Page 233: In Exercise 1.13.39, the factor of |x-y|^2 in the definition of K_t should just be |x|^2. In Exercise 1.13.40, “wave equation -\partial_{tt}u+\Delta u” should be “wave equation -\partial_{tt}u+\Delta u=0″, and “Schwartz functions f” should be “Schwartz functions f,g“.
  • Page 238:  In the display before Remark 1.14.1, the \partial_x symbols should be replaced by \partial x.  Similarly for the definition of L after Exercise 1.14.3.  In the definition of C^k({\bf R}^d), “derivatives of order k” should be “derivatives of order up to k“.
  • Page 239: In Exercise 1.14.3, C^\infty_c should be C^k, and N should be |\xi|.
  • Page 240: After Exercise 1.14.9, “C^{k,\alpha}({\bf R}^d) is contained in C^{k,\beta}({\bf R}^d)” should be “C^{k,\alpha}({\bf R}^d) contains C^{k,\beta}({\bf R}^d) for \alpha \leq \beta“.
  • Page 241: In Exercise 1.14.12(2), the additional hypothesis that k \leq l is missing.
  • Page 242: Exercise 1.14.13 is incorrect and should be replaced by the following: “Let {k \geq 0} and {0 \leq \alpha' < \alpha \leq 1}.  Show that {C^\infty_c({\bf R}^d)} is a dense subset of {C^{k,\alpha}_0({\bf R}^d)} if one places the {C^{k,\alpha'}} topology on the latter space. (Hint: To approximate a compactly supported {C^{k,\alpha}} function by a {C^\infty_c} one, convolve with a smooth, compactly supported approximation to the identity.) What happens in the endpoint case {\alpha=\alpha'}? “
  • Page 242: In Exercise 1.14.14, the signs are reversed in the formulae for u(x) and for \frac{\partial u}{\partial x_j} (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 \frac{\partial^2 u}{\partial x_i \partial x_j}(x) has both a reversed sign and a missing term; it should be\frac{1}{3} \delta_{ij} f(x) - \frac{1}{4\pi} \lim_{\varepsilon \to 0} \int_{|x-y| \geq \varepsilon} [ \frac{3(x_i-y_i)(x_j-y_j)}{|x-y|^5} - \frac{\delta_{ij}}{|x-y|^3}] f(y)\ dy.
  • Page 243: In Exercise 1.14.15, Kondrakov should be Kondrachov.
  • Page 244: In Exercise 1.14.16, \phi(x/R) \sin(\xi x) should be A \phi(x/R) \sin(\xi \cdot x).
  • Page 245: In Exercise 1.14.18, “is C^{k+1}({\bf R}^d)” should be “is the space C^{k}_0({\bf R}^d) of C^k functions such that the first k derivatives go to zero at infinity”.
  • 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”, and the annulus should be replaced with a ball.  In Exercise 1.4.22, the hypothesis that \Omega 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 L^p({\bf R}^d) norm should be L^p({\bf R}^{d-1}) instead.   (The d=1 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, \| f_i\|_{L^1({\bf R}^d)} should be \| f_i \|_{L^1({\bf R}^{d-1})}.
  • Page 251: in the third paragraph of Section 1.14.3, \frac{\partial f}{\partial x_j} f should be \frac{\partial f}{\partial x_j}.
  • Page 253: In Exercise 1.14.34, change “use Schur’s test” to “use the Cauchy-Schwarz inequality or Schur’s test”.  The hint in Exercise 1.14.35(i) will not work easily.  Instead, substitute: “First prove this when s is a non-negative integer using an argument similar to that in Exercise 1.14.12, then exploit duality to handle the case of negative integer s.  To handle the remaining cases, decompose the Fourier transform of f into annular regions of the form \{ \xi: 2^n \leq |\xi| \leq 2^{n+1}\} for n \geq 0, as well as the ball \{ \xi: |\xi| \leq 1 \}, and use the preceding cases to estimate the L^2 norm of the Fourier transform of fg these annular regions and on the ball.
  • Page 254: In the second display of Exercise 1.14.36, L(\xi_1,\ldots,\xi_d) should be l(\xi_1,\ldots,\xi_d).  In the third display, l(\xi) should be |l(\xi)|.
  • 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 d should be n throughout.
  • page 260: In the fourth bullet point, \overline{\hbox{dim}}_M should be \underline{\hbox{dim}}_M.  In the last two bullet points, \delta^{-n-\alpha - \varepsilon} should be \delta^{-n-\alpha+\varepsilon}.
  • Page 262: In (1.128) and (1.129), a logarithm is missing in the numerator.
  • Page 265: In the first paragraph on this page of Section 1.15.2, “measre” should be “measure”.
  • Page 266: In the paragraph before Exercise 1.15.9, “This quantity is increasing in r” should be “This quantity is decreasing in r“.
  • Page 267: In the proof of Proposition 1.15.3, A_l should be A_{1/l} in the definition of F_l.
  • Page 268: In the end of the proof of Lemma 1.15.4, \frac{\varepsilon}{k r^n} should be -\frac{\varepsilon}{kr^n}, and the inequalities c \geq \frac{1}{\omega_n} and c \leq \frac{1}{\omega_n} should be swapped.
  • Page 269: In Exercise 1.15.13, 0 \leq d < d' should be 0 \leq d' < d.
  • Page 271: In the proof of Lemma 1.15.7, “Huausdorff” should be “Hausdorff”.  In the second bullet point, \sum_{i=1}^k should be \sum_{i=1}^\infty.  In Exercise 1.15.17, “compact support there” should be “compact support and there”, and “dimension at most d” should be “dimension at least d” (two occurrences); similarly at the end of Lemma 1.15.7.
  • Page 273: In Exercise 1.15.20, the measure \mu should not just be compactly supported, but should be supported in E.
  • Page 288: In Lemma 2.2.18, replace the parenthetical with “the existence of which follows from the fact that G 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 K if necessary, we can take v to be one of the v_k.”
  • Page 291: In the proof of Corollary 2.2.26, SO(3) \backslash E should be S^2 \backslash E.
  • Page 293: In axiom (ii) at the start of 2.3, the second A \cup  (B \cap C) should be (A \cap B) \cup (A \cap C).
  • Page 303: In (2.14), x \in X should be x \in A.  In Exercise 2.4.2, Y should be y.
  • Page 305: In Proposition 2.4.13, “X and Y” should be “X to Y“.
  • 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, \{\beta,\{\beta\}\} should be \beta \cup \{\beta\}.
  • Page 311:In Definition 2.5.1, \pi should map from \overline{X} to \overline{X}', and \iota = \pi \circ \iota' should be \iota' = \pi \circ \iota.
  • Page 317: in the second display after (2.17), the second dx should be d\xi.
  • Page 319: In the last paragraph of 2.6.1, the term e^{-i\varepsilon e^{i\varepsilon} z^{2+\varepsilon}} should be e^{i\varepsilon e^{i\varepsilon} z^{2+\varepsilon}}.
  • Page 320: The factors of \pi e/a should be 2 \pi/ea, and \pi/ea should be 2 \pi e / a.
  • Page 334: In the proof of (i) implies (ii) in Theorem 2.8.1, \|\nu -\tau_x \nu\|_{L^1(G)} > \varepsilon should be \sup_{x \in S} \|\nu -\tau_x \nu\|_{L^1(G)} > \varepsilon, and “and all x \in S” should be omitted.

Errata for Vol. II:

  • Page 12: “a random complex number in {\bf C}” should be “a random complex vector in {\bf C}^n“.
  • Page 27: In the proof of Theorem 1.4.8, \bigcup_{\phi \in \Gamma'} F_\phi should be \bigcap_{\phi \in \Gamma'} F_\phi.
  • Page ???: In Proposition 1.5.6, the dimension should be n rather than d.
  • 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 n rather than d. After that proposition, insert “and dividing into whether {\bf E} f(X) is larger than or smaller than t/2, and noting also that the claim is trivial for t small” before “…”.
  • Page 41: prefactor \pi/2 should instead be 2/\pi (and conversely, 2/\pi should be \pi/2), and the final bound of \exp(Ct) should instead be \exp(Ct^2).
  • 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): F_p[X] should be ({\bf Z}/N{\bf Z})[X].
  • Page 90, in the display before (1.61): the final O(1) should be O(\sqrt{x}).
  • Page ???: In Section 1.10, “the bits of A on the support of s” should be “the bits of A' on the support of s“.
  • 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, g_{[[c,b],c]} should be added at the end,  and similarly for the first display on the next page; in the display after that, g_{[[c,b],c]}^{n_{[[c,b],c]}} should also be appended.
  • Page 310: “all group elements t \in K” should be “all group elements t \in H“.

Thanks to Marcelo Aguiar, Adam Azzam, Ravi Bajaj, 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, Cole Grunniger, Robert Hannah, Matthias Hübner, Joe Hughes, Ahmad Hussein, Ivo, Sune Kristian Jakobsen, Dominik Juestel, Zane Li, Dirk Lorenz, Martin Los, Cao Lu, Freddie Manners, Ian Martin, Ricardo Minares, Alexey Muranov, Sujit Nair, Nick, Seungly Oh, John Pearson, Qiang, Yudong Qiu, Rex, James Robinson, Agus Seonjaya, Lior Silberman, August Sonne, Dan Stroock, Jon Susice, Sunting, Yotas Trejos, Bruce Tu, Kent Van Vels, Anthony Verbitsky, Gerd Wachsmuth, Ben Wallis, Joshua Wilson,  wilsonofgordon, xuhmath, Sunting Yan, and Yaoliang  for corrections.