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.)
p. 6In 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”.
p. 7In Example 1.1.14, $\frac{1}{|E|}$ should be $\frac{1}{|X|}$.
p. 8In the first paragraph of the proof of Theorem 1.1.17, “see that if” should just be “see that”.
p. 9In 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).
p. 15In (1.5), $d\mu$ should be $dm$.
p. 17Before (1.9), “$c>0$ should be $c \geq 0$“.
p. 18After Definition 1.2.1: “one of the signed parts” should be “one of the unsigned parts”.
p. 20In 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.
p. 22After Remark 1.2.8, “”singular continuous”” should be “”absolutely continuous””.
p. ?Around (1.13)-(1.14) and preceding: most of the signs and inequality signs here need to be reversed.
p. 33In (1.18), $f^r$ should be $|f|^r$.
p. ?in the paragraph before Section 1.3.1, “Frechet” should be “Fréchet”.
p. 34In Exercise 1.3.2 (iv), in the $0 < p < 1$ case the additional hypothesis that $f, g$ are non-negative is required.
p. 36After 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 $L^p$.” After Remark 1.3.6: Exercise 1.3.5 should be Exercise 1.3.7.
p. ?In the proof of Proposition 1.3.8, “the space … are” should be “the space … is” (two occurrences).
p. 39In Exercise 1.3.10, $f^p$ and $g^q$ should be $|f|^p$ and $|g|^q$ respectively.
p. 41In 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”. “$n \to \infty$” should be “$N\to \infty$“.
p. 42In the second to last line, $m=1$ should be $g=1$.
p. 43At the end of the first sentence: $E'$ should be $E$.
p. 46In equation (1.38), the dot product should be an inner product.
p. 50In Exercise 1.4.6, “this inequality” should be “this equality”, and “non-empty finite measure” should be “non-zero finite measure”.
p. 51In Exercise 1.4.8, a right parenthesis is missing after “of $\overline{V}$ and $\overline{V}'$“.
p. ?In Example 1.4.10, $l^2$ should be $\ell^2$. (Similarly with $l^1$ and $l^\infty$ at several places in Section 1.5.)
p. ?In Exercise 1.4.12, “Then” should be “Show that”.
p. 53In Remark 1.4.15, “next set of notes” should be “next section”.
p. 54In the second and third bullets of Exercise 1.4.16, $T^\dagger \mathrm{T}$ should be $T^\dagger T$.
p. 55In 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”.
p. ?In the proof of Proposition 1.4.18, “contradict orthogonality” should be “contradict maximality”.
p. 61When 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”.
p. 62In Exercise 1.5.7, $g \in \bar{H}$ should be $\bar{g} \in \bar{H}$.
p. 63In Exercise 1.5.10, “$n \times m$” should be “$m \times n$“. 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.
p. 64In the second display in the proof of Proposition 1.5.7, $\lambda(y') - \|y'+v\|_X$ should be $-\lambda(y') - \|y'+v\|_X$.
p. 65In the proof of the complex case of the Hahn-Banach theorem, $\tilde \lambda$ should take values in ${\mathbf C}$ rather than ${\mathbf R}$. In Exercise 1.5.13, $\to$ should be $\mapsto$ (and similarly for Remark 1.7.11).
p. 65In 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 $X^*$.
p. 66In 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 $X$.
p. 67In 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”.
p. 68In 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”.
p. ?In Exercise 1.6.5, “Then” should be “Show that”. In Exercise 1.6.6, delete the requirement that $f,g$ be continuous.
p. 72In Definition 1.6.6, after introducing the notion of an isolated point, add “(a point which is open)”.
p. 74In the proof of (ii) implying (i) in Theorem 1.6.8, a closed parenthesis should be added after “converging to $x$“.
p. 79In Example 1.6.21, $n \to \infty$ should be $n \to +\infty$.
p. 80In 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 $(f,g)(z) := (f(z),g(z))$“.
p. ?In Example 1.6.27, the reference to Example 1.6.13 should instead be to Example 1.6.12.
p. 81In Definition 1.6.29, “partially ordered set” should be “pre-ordered set”. To conform to usual notational conventions, the partial order should be denoted $\leq$ rather than $<$.
p. 87In the last sentence of the proof of the Baire category theorem, $B$ should be $B(x_0,r_0)$.
p. ?In the second sentence of Section 1.7.2, the comma should be after “from” rather than before.
p. 89In 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”.
p. 90In 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.
p. 92In 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 $4n (1 + \frac{\|f_0\|_Y}{r})$.
p. 93In 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”.
p. 94In Exercise 1.7.7, “we can ensure that” may be clarified to “the above statement remains true if we impose the additional condition that”.
p. 95In 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”.
p. 97In 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. The reference to (1.103) can be replaced with Section 1.12.2.
p. 98In the proof of Theorem 1.7.22, all occurrences of $\mu$ should be replaced by $\sigma$.
p. 100“Phi, Isett” should be “Phil Isett”.
p. ?In Exercise 1.8.3, $X$ should be $X = (X,{\mathcal F})$.
p. 103Before Exercise 1.8.4, in the definition of an ultrafilter, $E \in X$ should be $E \subset X$.
p. 104In Exercise 1.8.8, “basis” should be “base”.
p. 105In Exercise 1.8.11, “every sufficiently large $x_n$” should be “every sufficiently large $n$“.
p. ?In Exercise 1.8.14, $X$ should be assumed to be non-empty, and Exercise 1.6.10 should be Example 1.6.21.
p. ?In the proof of Proposition 1.8.12, $x_2$ should belong to $X_2$, not $X_1$.
p. 111In Exercise 1.8.21, the optional fifth part is incorrect and should be deleted.
p. 112In the last bullet point of Definition 1.8.18, $x',x \in x$ should be $x',x \in X$.
p. ?In Exercise 1.8.23, a “Show that” is missing, and “upsets” should be “subsets”. After Exercise 1.8.26, “Note that if” should just be “Note that”.
p. 113In Example 1.8.21, “$X=Y={\bf R}$” should be “$X = {\bf R} \backslash \{0\}$ and $Y = {\bf R}$“. In Example 1.8.22, equicontinuity can be upgraded to uniform equicontinuity.
p. 114$f_\alpha(x_n)$ should be $f_n(x_n)$.
p. 115In the first line, $F_n$ should be $F_{n_j}$.
p. 118In Definition 1.9.1, “is continuous” should be “are continuous”.
p. ?In Exercise 1.9.2, a “Show that” is missing.
p. 119At 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. In Remark 1.9.7, “lectures” should be “sections”, and Exercise 1.9.6 should be Example 1.9.6.
p. 120In 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.
p. ?Before Remark 1.9.11, “viewed as a function” should be “viewed as a sequence of functions”.
p. 123In 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.
p. 124In Remark 1.9.15, $\ell^\infty({\bf N})$ should be $\ell^\infty({\bf N})^*$.
p. 125In Exercise 1.9.19, the third item is incorrect and should be deleted. “As $n \to \infty$” should be “As $N \to \infty$“.
p. 127In Remark 1.9.17, “proper chain” should be “proper well-ordered chain” throughout.
p. ?In Example 1.9.21, “translation operator by” should be “translation operator defined by”.
p. 129In Exercise 1.9.25, the first sentence can be deleted (also the word “throughout” should be appended to the last sentence).
p. 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.
p. 134“Gelfand-Neimark” should be “Gelfand-Naimark”.
p. 136In Exercise 1.10.4, “on this finite set” should be “on this countable set”. In Definition 1.10.2, ${\bf R}^+$ should be $[0,+\infty)$.
p. 137In 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. To retain vanishing at infinity, one can either work using the one-point compactification, or decompose a $C_0$ function as an absolutely convergent series of $C_c$ functions.)”. This exercise should thus also be moved to after the Tietze extension theorem.
p. 139In Remark 1.10.7, the final $K$ should be an $X$.
p. ?In Lemma 1.10.9, “belongs to” should be “intersects”.
p. 140In 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.
p. 141In 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 $x$).”
p. 146After “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 $U$“.
p. ?In Exercise 1.10.14, add that the lower semicontinuous function $g$ is permitted to take infinite values.
p. 148In 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 $f \in C_c(X \to {\bf R})$.”
p. 149After 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).
p. ?Before Exercise 1.10.23, “algebras that does” -> “algebras that do”. After this exercise, “axiom” should be “hypothesis”.
p. 152In Exercise 1.10.25, $[{\bf R},{\bf Z}]$ should be ${\bf R}/{\bf Z}$. In Exercise 1.10.24, “Then” should be “Show that”.
p. 154In 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.”.)
p. 155In Definition 1.10.25, “$C^*$ algebra” is missing a hyphen. In Exercise 1.10.35, insert “when $x$ is self-adjoint” after the second display.
p. ?After Example 1.11.1, a superfluous “with $\theta$” can be deleted.
p. 160in the last paragraph, the closing parenthesis after (1.80) should instead be after “convex function of $\theta$“.
p. 161In (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.
p. 162In Exercise 1.11.4, $f(0+it)$ and $f(1+it)$ should be $|f(0+it)|$ and $|f(1+it)|$ respectively, and similarly for $f(\sigma+it)$.
p. 164After (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 $|f(x)|$.
p. 166In the fourth display, the final $\|f\|_{L^{p,\infty}(X)}$ should be $\|g\|_{L^{p,\infty}(X)}$. “finite $L^{p,\infty}(X)$” should be “finite $L^{p,\infty}(X)$ quasi-norm”. 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.)
p. 167In 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.
p. 168In Exercise 1.11.10, $f^*$ should take values in $[0,+\infty]$ rather than ${\bf R}^+$, and $t \geq 0$ should be replaced by $t>0$. Instead of $f$ be finite almost everywhere, impose that $f \in L^p(X)$ for some finite $p$.
p. 169In 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.
p. 170In Remark 1.11.8, the second semicolon should be a comma.
p. 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.”.
p. 172In 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).
p. 173Just 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.
p. 174-175All occurrences of “(1.91), (1.94)” should be “(1.93), (1.94)”.
p. 176In 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 $p_0 \neq p_1$.
p. 177In Exercise 1.11.21, $kO(1)$ should be $k+O(1)$.
p. ?In Exercise 1.11.22, add the hypothesis $p \leq q$.
p. 180In Exercise 1.11.24, $x^{r'}$ should be $y^{r'}$.
p. 181In 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.) In the prologue, “two functions $f$” should be “two functions $f,g$“.
p. 187In 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 $\tilde L^\infty(G)$.”. In Exercise 1.12.6, “to be the an” should be “to be an”.
p. 188In 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.]
p. 189In Exercise 1.12.7(g), “range in” should be “takes values in”.
p. 190Exercise 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”.
p. 194In the “Unitarity” component of Corollary 1.12.5, “Thus the” should simply be “The”.
p. 196In 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 $\frac{\sin(\pi N x)}{\sin(\pi x)}$.
p. 197In Exercise 1.12.24, all occurrences of $x_n$ should be $x_d$.
p. 198In 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.
p. ?In the definition of the Fejer kernel, 1/n should be 1/N. In the subsequent exercise, D_N should be S_N.
p. ?After (1.107), “propeties” should be “properties”.
p. 200In Exercise 1.12.33, $f(x_j g)$ should be $f * (x_j g)$.
p. ?In Theorem 1.12.10, ${\mathcal S}$ should be ${\mathcal S}({\mathbf R}^d)$.
p. 201In 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.
p. 203In Exercise 1.12.37, (1.103) should be (1.111). The minus signs in the entropic uncertainty principle should be plus signs, and the improved constant should be $d\log(e/2)$ rather than $d\log(2e)$.
p. 209In the display discussing the FFT, the summation should be over $x$ rather than $n$. In the discussion of the $z$-transform $z$ should be $e^{2\pi i x}$ rather than $e^{2\pi i nx}$.
p. 214The 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, delete the sentence before “Because of this” and replace the next sentence with “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.”
p. 215Exercise 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.
p. 216The 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.
p. 217In 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)”.
p. ?In Exercise 1.13.9, change all “converge” to “converges” (and “they” to “it”).
p. 220After (1.114), $\lambda h$ should be $\lambda * h$.
p. 224In the third bullet point of Exercise 1.13.25, “some compactly supported distributions” should be “some compactly supported distributions $\rho,\lambda$“.
p. 225In 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.
p. 227In Remark 1.13.10, it is the word “both” that should be italicised, rather than “growth”.
p. 230In the final display, $t^d$ should be $t^{-d}$.
p. ?In Exercise 1.13.26: In the second part, the distributions of the left-hand side should be composed rather than multiplied; and in the third part, the conclusion is to show that $\lambda$ is a linear combination of the Dirac delta and $\hbox{p.v.} 1/x$.
p. ?In Exercise 1.13.31, “is the … function” should be “are the … functions”.
p. 232In 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 $|x|^2$.
p. 233In 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 $f,g$“.
p. 238In 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 $k$“.
p. 239In Exercise 1.14.3, $C^\infty_c$ should be $C^k$, and $N$ should be $|\xi|$.
p. 240After 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 $\alpha \leq \beta$“. Also, $C^{k+1}$ should be $C^{k+1}({\bf R}^d)$.
p. ?In Exercise 1.14.10, $0 \leq \alpha \leq 1$ should be $0 < \alpha < 1$.
p. 241In Exercise 1.14.12(2), the additional hypothesis that $k \leq l$ is missing.
p. 242Exercise 1.14.13 is incorrect and should be replaced by the following: “Let ${k \geq 0}$ and ${0 \leq \alpha' 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 ${\alpha=\alpha'}$? “
p. 242In 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$\frac{1}{3} \delta_{ij} f(x) - \frac{1}{4\pi} \lim_{\varepsilon \to 0} \int_{|x-y| \geq \varepsilon} [ \frac{3(x_i-y_i)(x_j-y_j)}{|x-y|^5} - \frac{\delta_{ij}}{|x-y|^3}] f(y)\ dy$.
p. 243In Exercise 1.14.15, Kondrakov should be Kondrachov.
p. 244In Exercise 1.14.16, $\phi(x/R) \sin(\xi x)$ should be $A \phi(x/R) \sin(\xi \cdot x)$.
p. 245In 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”.
p. 246In Theorem 1.14.7, “encluding” should be “excluding”.
p. 248In 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.
p. 249Before 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 $\| f_i \|_{L^1({\bf R}^{d-1})}$.
p. ?Before Exercise 1.14.29, “the variable of independent variable” should just be “the independent variable”.
p. ?In Exercise 1.14.33, $\tau_x(y)$ should be $\tau_x f(y)$, and “previous exercise” should be “previous part”.
p. 251in the third paragraph of Section 1.14.3, $\frac{\partial f}{\partial x_j} f$ should be $\frac{\partial f}{\partial x_j}$.
p. 253In 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.
p. 254In 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 $|l(\xi)|$. $\partial \xi_{j_1} \dots \partial \xi_{j_d}$ should be $\partial \xi_1^{j_1} \dots \partial \xi_d^{j_d}$, and similarly for $\xi_{j_1} \dots \xi_{j_d}$.
p. ?In Remark 1.14.10, “parabolic operators” should be “a parabolic operator”, and similarly for “hyperbolic operators”. “Controling” should be “Controlling”.
p. 258Before (1.125): “chain of maximal ideals” should be “chain of prime ideals”. Bougliand should be Bouligand.
p. 259In Section 1.15.1, “k-dimensional subspace” should be “d-dimensional subspace”, and in the second paragraph $d$ should be $n$ throughout.
p. ?After Exercise 1.15.3, “$\delta$-metric entropy” should be “$\delta$-metric entropy of $E$“.
p. 260In 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 $\delta^{-n-\alpha+\varepsilon}$.
p. ?Before Exercise 1.15.6, $d = \mathrm{dim}_H$ should be $d = \mathrm{dim}_H(E)$.
p. 262In (1.128) and (1.129), a logarithm is missing in the numerator. In Exercise 1.15.7, the phrase “and any $x_0 \in X$” should be deleted.
p. ?In the proof of Lemma 1.15.7, “Huausdorff” should be “Hausdorff”.
p. 265In the first paragraph on this page of Section 1.15.2, “measre” should be “measure”.
p. 266In the paragraph before Exercise 1.15.9, “This quantity is increasing in $r$” should be “This quantity is decreasing in $r$“.
p. 267In the proof of Proposition 1.15.3, $A_l$ should be $A_{1/l}$ in the definition of $F_l$.
p. 268In 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.
p. 269In Exercise 1.15.13, $0 \leq d < d'$ should be $0 \leq d' < d$.
p. 271In 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.
p. 273In Exercise 1.15.20, the measure $\mu$ should not just be compactly supported, but should be supported in $E$; and the integrals should be on ${\bf R}^n$ rather than ${\bf R}^d$. Before this exercise, the accent on Caratheodory is in the wrong location.
p. ?Before Theorem 1.8.4, $\{0,1\}^\infty$ should be $\{0,1\}^{\bf N}$.
p. ?In Definition 1.8.18, $x', x \in x$ should be $x',x \in X$.
p. ?In the final paragraph of the proof of Theorem 1.8.23, “$f_\alpha(x_m)$ takes” should be “the $f_n(x_m)$ take”, and in the final sentence, $f_n, F_n$ should be $f_{n_j}, F_{n_j}$.
p. 286In the paragraph before 2.2.7, a right parenthesis is missing after “are amenable”.
p. 287In the first paragraph of Section 2.2.2, G should be in math mode.
p. 288In 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.
p. 289In 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 $v_k$.”
p. 290add a right parenthesis to the end of Exercise 2.2.13.
p. 291In the proof of Corollary 2.2.26, $SO(3) \backslash E$ should be $S^2 \backslash E$.
p. 293In axiom (ii) at the start of 2.3, the second $A \cup (B \cap C)$ should be $(A \cap B) \cup (A \cap C)$. “Because the ${\mathcal B}$” should just be “Because ${\mathcal B}$“.
p. 296In the proof of Theorem 2.3.4, one of the B’s (in “map B to 1”) should be in math mode.
p. ?Before Remark 2.3.5, “indictator” should be “indicator”.
p. 298In the proof of Theorem 2.3.10, omit “one” in “Boolean algebra isomorphism one instead”. A right parenthesis was omitted after “clopen algebra of $X$“. $B_1,B_2,\cdots$ should be $B_1,B_2,\dots$ (two occurrences).
p. ?In Exercise 2.3.3, “quotieting” should be “quotienting”, and “notion… is” should be “notions… are”.
p. 299In Remark 2.3.11: N should be in math mode in “empty interior is in N”.
p. 303In (2.14), $x \in X$ should be $x \in A$. In Exercise 2.4.2, $Y$ should be $y$.
p. 304In Exercise 2.4.4, $\mapsto$ should be $\to$.
p. 305In Proposition 2.4.13, “$X$ and $Y$” should be “$X$ to $Y$“.
p. 307In Remark 2.4.22, “Exercises 2.4.3 and 2.4.4” should be “Examples 2.4.6 and 2.4.7”.
p. 308In Exercise 2.4.10, $\{\beta,\{\beta\}\}$ should be $\beta \cup \{\beta\}$.
p. 311In Definition 2.5.1, $\pi$ should map from $\overline{X}$ to $\overline{X}'$, and $\iota = \pi \circ \iota'$ should be $\iota' = \pi \circ \iota$.
p. ?In Example 2.5.2, ${\mathbf R}^2$ should be ${\mathbf R}^3$.
p. ?In Exercise 2.5.4, Theorem 1.2.2 should be Theorem 2.3.4.
p. 317in the second display after (2.17), the second $dx$ should be $d\xi$.
p. 319In the last paragraph of 2.6.1, the term $e^{-i\varepsilon e^{i\varepsilon} z^{2+\varepsilon}}$ should be $e^{i\varepsilon e^{i\varepsilon} z^{2+\varepsilon}}$.
p. 320The factors of $\pi e/a$ should be $2 \pi/ea$, and $\pi/ea$ should be $2 \pi e / a$.
p. 333“whenever the right-hand side is convergent” should be “whenever the right-hand side is absolutely convergent”.
p. 334In 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.
p. ?In the proof of Proposition 2.8.5, the definition of $A$ should involve $E_1$ rather than $E_{-1}$.
Volume II
18 corrections · 5 awaiting a page number
p. 12“a random complex number in ${\bf C}$” should be “a random complex vector in ${\bf C}^n$“.
p. ?in the proof of Corollary 1.2.7, all occurrences of $R_j \cdot P$ should be $R_j \circ P$.
p. ?Just before Theorem 1.4.5, $\Gamma$ should contain $\neg A_1$ rather than $A_1$.
p. 27In the proof of Theorem 1.4.8, $\bigcup_{\phi \in \Gamma'} F_\phi$ should be $\bigcap_{\phi \in \Gamma'} F_\phi$.
p. ?In Proposition 1.5.6, the dimension should be $n$ rather than $d$.
p. 40In 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 “…”.
p. 41prefactor $\pi/2$ should instead be $2/\pi$ (and conversely, $2/\pi$ should be $\pi/2$), and the final bound of $\exp(Ct)$ should instead be $\exp(Ct^2)$.
p. 43In 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.
p. 5430.7% should be 30.1% .
p. 82before (1.53): $F_p[X]$ should be $({\bf Z}/N{\bf Z})[X]$.
p. 90in the display before (1.61): the final $O(1)$ should be $O(\sqrt{x})$.
p. ?In Section 1.10, “the bits of $A$ on the support of $s$” should be “the bits of $A'$ on the support of $s$“.
p. 112In Remark 1.15.20, “incompleteness theory” should be “incompleteness theorem”.
p. 177After (2.19), “existence of a quadratic residue” should be “existence of a quadratic nonresidue”.
p. ?In the paragraph after (2.27), $x^p=p$ should be $x^p=x$.
p. 235The 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.
p. 238In 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.
p. 310“all group elements $t \in K$” should be “all group elements $t \in H$“.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.