Last updated: May 17, 2013

An epsilon of room: pages from year 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.

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 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”.   In Corollary 1.2.5, replace $f \in L^1(X,dm)$ with “$f: X \to {\bf R}$ measurable”.
• 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 $\mu_{\overline{g}}$.
• 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 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 ofof isomorphism.)
• 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 Exercise 1.5.14, $\|\lambda\|_{X^*}=1$ should be $\|\lambda\|_{X^*} \leq 1$.   In the proof of Hahn-Banach in the complex case, “$\tilde \rho$ has norm at most 1″ should be ”$\tilde \lambda$ has norm at most 1″.
• Page 67: In the proof of Theorem 1.5.13, $\|x\|$ should be $\|x\|_X$.
• Page 68: In Remark 1.5.15, add “, with $\lambda$ is now required to be non-zero”.
• 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 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.  is the Lipschitz norm, but the reader is invited to experiment with other choices as well.
• Page 92: In the proof of Theorem 1.7.12, $nr+n\varepsilon$ should be $n\|f_0\|_Y + nr + n\varepsilon$, with similar changes in the next ; also, $\frac{5}{2} n$ and $latex$ in the following paragraphs need to be replaced by some other constants that are messier, but still independent of $f$.)
• Page 93: In Exercise 1.7.6, the Hahn-Banach theorem is actually not necessary for this exercise.  It should also be emphasised that the notion of isomorphism here is the Banach space notion (see Section 1.5.1), and not the Hilbert space notion of isometric isomorphism from after Exercise 1.4.1.
• Page 95: In Example 1.7.18, both occurrences of $c_0({\bf N})$ should be $c_c({\bf N})$.
• Page 98: In the proof of Theorem 1.7.22, all occurrences of $\mu$ should be replaced by $\sigma$.
• 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, in the optional fifth part, the hypothesis “Let $X$ be a first-countable topological space” is missing.
• Page 113: In Example 1.8.21, “$X=Y={\bf R}$” should be “$X = {\bf R} \backslash \{0\}$ and $Y = {\bf R}$“.
• Page 119: 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 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 125: In Exercise 1.9.19, the third item is incorrect and should be deleted.
• Page 127: In Definition 2.6.16, it should be stressed that the fibre metrics $d_y$ are compatible with (i.e. generate) the topology on the fibres inherited from the full space.  (More generally, in this text, when we refer to a metric on a topological space, it should be understood that that metric generates the topology of that space unless otherwise specified.)
• Page 128: In the first paragraph, $f(y_0)$ should be $f(x_0)$.
• Page 132: In Lemma 2.6.30, $Y \times_\sigma ({\bf R}/{\bf Z})$ should be $Y \times_\sigma ({\bf R}/{\bf Z})^d$.
• 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”.
• 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).
• 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”.  Exercise 1.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 $f \in C_c(X \to {\bf R})$.”
• Page 152: In Exercise 1.10.25, $[{\bf R},{\bf Z}]$ should be ${\bf R}/{\bf Z}$ .
• Page 154: In Exercise 1.10.30, the phrase “on compact subsets of ${\bf R}$” is redundant and can be deleted.
• 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, 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 166: In Example 1.11.6, the first $q>p$ should be $q.  In Exercise 1.11.7, the dyadic Lorentz norm $\|f\|_{L^{p,q}(X)}$ defined in the exercise is not quite a quasi-norm, because the homogeneity axiom $\|\lambda f \|_{L^{p,q}(X)} = |\lambda| \|f\|_{L^{p,q}(X)}$ is not satisfied.  Instead, weakened version of the quasi-norm axioms in which the homogeneity axiom is replaced by a quasi-homogeneity axiom $c|\lambda|\|f\|_{L^{p,q}(X)}$ $\leq\|\lambda f \|_{L^{p,q}(X)}$ $\leq C |\lambda|$ $\| f \|_{L^{p,q}(X)}$ for some $C,c>0$ 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 171: “it is in fact convex in all of $[0,+\infty)^2$” should read “it is also convex in the triangular region $\{ (1/p,1/q) \in [0,+\infty)^2: p \leq q \}$“.
• Page 173: Just before Theorem 1.11.10, Marcinkeiwicz should be Marcinkiewicz.
• Page 176: In the first display, $2^{n\alpha q_\theta - mp_\theta}$ should be $2^{(n_\alpha q_\theta - m p_\theta) x_i}$.
• Page 177: In Exercise 1.11.21, $kO(1)$ should be $k+O(1)$.
• 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 $d\mu(x)$ .
• 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).
• 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 198: In the display in Exercise 1.12.25, ${\bf R}^n$ should be ${\bf R}^d$.
• 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 215: Exercise 1.13.3(iii) is incorrect and should be replaced with the following: “(iii) Despite this, show that a set $E \subset C^\infty_c({\bf R}^d)$ is (topologically) closed if and only if it is sequentially closed (i.e. whenever $f_n$ is a sequence in $E$ that converges in $C^\infty_c({\bf R}^d)$ to a limit $f$, then $f \in E$).  Conclude that a map $T: C^\infty_c({\bf R}^d) \to X$ from the space of test functions into a topological 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 $X$. Thus while first countability fails for $C^\infty_c({\bf R}^d)$, we have a servicable substitute for this property.”.  Also add “(iv) 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 $E$.  Why does this not contradict (iii)?”.
• 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.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 $n$“.
• 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 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).
• 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.
• 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 contained in $C^{k+1}({\bf R}^d)$“.
• Page 249: Before Exercise 1.14.23, “, which we will do in later notes” should be “; see Exercise 1.15.23″.
• Page 258: Before (1.125): “chain of maximal ideals” should be “chain of prime ideals”.
• Page 259: In Section 1.15.1, “k-dimensional subspace” should be “d-dimensional subspace”.
• 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, $\frac{\varepsilon}{k r^n}$ 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 271: In the proof of Lemma 1.15.7, “Huausdorff” should be “Hausdorff”.
• Page 303: In (2.14), $x \in X$ should be $x \in A$.
• Page 308: In Exercise 2.4.10, $\{\beta,\{\beta\}\}$ should be $\beta \cup \{\beta\}$.
• 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 40: In the statement of Proposition 1.5.7, a factor ofthe right hand side. 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 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 Adam Azzam, Farzin Barakat, Lucas Braune, Sean Eberhard, Stephen Ge, Wengyin Gan, Joe Hughes, Sune Kristian Jakobsen, Dirk Lorenz, Martin Los, Freddie Manners, Ian Martin, Ricardo Minares, Alexey Muranov, Sujit Nair, Seungly Oh, John Pearson, Qiang, Rex, Lior Silberman, Dan Stroock, Jon Susice, Anthony Verbitsky, Ben Wallis, Joshua Wilson,  xuhmath and Yaoliang  for corrections.