Analysis II

Last updated Mar 5, 2012

Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 274 pages. ISBN 81-85931-62-3

This is basically an expanded and cleaned up version of my lecture notes for Math 131B. In the US, it is available through the American Mathematical Society. It is part of a two-volume series; here is my page for Volume I.

— Errata to the first edition —

p. 392, example 12.1.7: $5+2=7$ should be $3+4=7$ .
p. 393, example 12.1.9: $\sup(5,2)=7$ should be $\sup(3,4)=4$ .
p. 394, example 12.1.13: (iii) should be (c).
p. 403, example 12.2.13: delete the redundant “, but not the other”.
p. 404, line 4: “neither open and closed” should be “neither open nor closed”.
p. 415, line 3: $\alpha \in A$ should be $\alpha \in I$ .
p. 416, line 11: “ $k \geq j$ ” should be “ $k > j$ “.
p. 419, line -2: In Exercise 12.5.15, = should be $\neq$ . Also, “that by counterexample” should be “by counterexample that”
p. 426, Exercise 13.2.9: $X$ should be ${\Bbb R}$ throughout. Also, the definition of limsup and liminf for functions has not been given; it can be reviewed here, e.g. by inserting “where we define $\limsup_{x \to x_0} f(x) := \inf_{r>0} \sup_{x: |x-x_0| \leq r} f(x)$ and $\liminf_{x \to x_0} f(x) := \sup_{r>0} \inf_{x: |x-x_0| \leq r} f(x)$ .”
p. 435, Definition 13.5.6: “metric space” should be “topological space”.
p. 438, Exercise 13.5.9: One needs to assume as an additional hypothesis that X is first countable, which means that for every x in X there exists a countable sequence V_n of neighborhoods of x, such that every neighbourhood of x contains one of the V_n.
p. 452, Exercise 14.3.6: “Propositoin” should be “Proposition”.
p. 452, Exercise 14.3.8: “ $x \in {\Bbb R}$ ” should be “ $x \in X$ “.
p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
p. 459: In line 11, $2 \epsilon$ should be $\epsilon$ .
p. 464: ) missing at the end of Exercise 14.7.2. An additional exercise, Exercise 14.7.3 is missing; it should state “Prove Corollary 14.7.3.”.
p. 466: Exercise 14.8.8 should be Exercise 14.8.2.
p. 467: Exercise 14.8.11 should be Exercise 14.8.4.
p. 469: “Limits of integration” should be “Limits of summation”. In Lemma 14.8.14, $3M+2\delta$ should be $1+4M$ , and Exercise 14.8.14 should be Exercise 14.8.6.
p. 470: Exercise 14.8.15 should be Exercise 14.8.7. Exercise 14.8.16 should refer to a (currently non-existent) Exercise 14.8.9, which of course would be to prove Lemma 14.8.16.
p. 471: At the end of the proof of Corollary 14.8.19, $|P(y)-g(y)| \leq \varepsilon$ should be $|P(y)-f(y)| \leq \varepsilon$ .
p. 472: In Exercise 14.8.2(c), Lemma 14.8.2 should be Lemma 14.8.8.
p. 477: In Exercise 15.1.1(e), Corollary 14.8.18 should be Corollary 14.6.2.
p. 478: In Example 15.2.2, $\frac{1}{x-1}$ should be $\frac{1}{1-x}$ .
p. 482: In Exercise 15.2.5, the $(x-b)^n$ on the right-hand side should be $(x-b)^m$ .
p. 486: In second and third display, y should be in ${}[0,1]$ rather than $(-1,1)$ .
p. 493: In Exercise 15.5.4, $x < 0$ should be $x \leq 0$ .
p. 501: In Theorem 17.7.2, “if $f(x_0)$ is not invertible” should be ”if $f'(x_0)$ is not invertible”.
p. 502: In Exercise 15.6.6, Lemma 15.6.6 should be Lemma 15.6.11.
p. 511: “Fourier… was, among other things, the governor of Egypt during the reign of Napoleon. After the Napoleonic wars, he returned to mathematics.” should be “Fourier… was, among other things, an administrator accompanying Napoleon on his invasion of Egypt, and then a Prefect in France during Napoleon’s reign.”
p. 556: In Theorem 17.5.4, f can take values in ${\Bbb R}^m$ and not just in $R$ ; insert the line “By working with one component of $f$ at a time, we may assume $m=1$ ” as the first line of the proof. Also, $f(x-x_0)$ should be $f(x+x_0)$ .
p. 557: In the second display, $-f(0)$ should be $+f(0)$ .
p. 560: In Exercise 17.6.1, add the hypothesis “and $f'$ is continuous” before “, show that $f$ is a strict contraction”.
p. 561: In Exercise 17.6.3, change “which is a strict contraction” to “such that $|f(x)-f(y)| < |x-y|$ for all distinct $x,y$ in ${}[a,b]$ “. In Exercise 17.6.8, $\max(c,c')$ should be $\min(c,c')$ .
p. 562: In Theorem 17.7.2, $T: E \to {\Bbb R}^n$ should be $f: E \to {\Bbb R}^n$ .
p. 565, line -7: $U$ should be $f^{-1}(B(0,r/2))$ rather than $f^{-1}(B(0,r))$ .
p. 570, first display: all partial derivatives should have a – sign (not just the first one). Last paragraph: “Thus $(x_1,\ldots,x_{n-1})$ lies in W” should be “Thus $(x_1,\ldots,x_{n-1})$ lies in U”.
p. 571, second display: add “ $=0$ ” at the end.
p. 584, Corollary 18.2.7: “ $x_i \in [a_i,b_i]$ ” should be “ $x_i \in (a_i,b_i)$ “.
p. 599, Definition 18.5.9: $(a,\infty)$ should be $(a,+\infty]$ .
p. 600: In Lemma 18.5.10, $f: \Omega \to {\Bbb R}$ should be $f: \Omega \to {\Bbb R}^*$ . In the second and fourth lines of the proof of this lemma, $(a,+\infty)$ should be $(a,+\infty]$ .
p. 616-617, Exercise 19.2.10: ${\Bbb R}$ should be ${}[0,1]$ throughout.

— Errata to the second edition —

p. 419, In Exercise 12.5.12(b), the phrase “with the Euclidean metric” should be deleted.
p.437: In Exercise 13.5.5, “there exist $a, b \leq X$ such that the “interval” $\{ y \in X: a < y < b \}$ ” should be replaced with “there exists a set $I$ which is an interval $\{y \in X: a < y < b\}$ for some $a, b \in X$ , a ray $\{y \in X: a < y \}$ for some $a \in A$ , the ray $\{ y \in X: y < b\}$ for some $a \in A$ , or the whole space $X$ , which”. In Exercises 13.5.6 and 13.5.7, “Hausdorff” should be “not Hausdorff”.
p. ???: In Exercise 14.1.5, $\lim_{y \to y_0; y \in f(E)} g(x)$ should be $\lim_{y \to y_0; y \in f(E)} g(y)$ , and $\lim_{x \to x_0; x \in E} g \circ f(x_0)$ should be $\lim_{x \to x_0; x \in E} g \circ f(x)$ .
p. ???: In Theorem 15.1.6(d), the summation should start from n=1 rather than n=0.
p. 478: Just before Definition 15.2.4, “for some $a \in {\ bf R}$ ” should be “for some $r > 0$ “.
p. 483: In Exercise 15.2.8(e), “ $d_m (x-b)^n$ ” should be “ $d_m (x-b)^m$ .
p. 485: $s_N$ in the third display and $s_0$ in the next line should be
$S_N$ and $S_0$ respectively.
p. 515: In the display before Definition 18.2.4, $j=\in J$ should be $j \in J$ . In Definition 18.2.4, $\sum_{j=1}^\infty$ should be $\sum_{j \in J}$ .
p. 531: In Exercise 16.5.4, $in \hat f(n)$ should be $2\pi i n \hat f(n)$ .
p. 536: In Example 17.1.7, $x \in {\Bbb R}$ should be $c \in {\Bbb R}$ .
p. 545: In Corollary 19.2.11, “non-negative functions” should be “non-negative measurable functions”.
p. 546: In Definition 17.3.7, $1 \leq j \leq m$ should be $1 \leq j \leq n$ , and $x_0+tv$ should be $x_0+te_j$ .
p. 548: In the definition of L in the proof of Theorem 17.3.8, m should be n.
p. 568: In the first paragraph, the function $f$ should be $g$ (for better compatibility with the discussion of the implicit function theorem).
p. 552: In Exercise 17.3.1, Exercise 17.1.3 should be Exercise 17.2.1.
p. 560: After “ $F$ does indeed map $B(0,r)$ to itself.”, add “The same argument shows that for a sufficiently small $\varepsilon > 0$ , $F$ maps the closed ball $\overline{B(0,r-\varepsilon)}$ to itself. After “ $F$ is a strict contraction”, add “on $B(0,r)$ , and hence on the complete space $\overline{B(0,r-\varepsilon)}$ “.
p.563, line 17: “ $f^{-1}(y)={\tilde f}^{-1}(y+f(x_0))$ ” should be “ $f^{-1}(y)={\tilde f}^{-1}(y-f(x_0))$ “.
p. 567: On the second to last line (in Section 17.8), $(0,1)$ should be $(-1,0)$ .
p.570, second line of the last paragraph, “U is open and contains $(y_1, \dots,y_{n-1},0)$ ” should be “U is open and contains $(y_1, \dots,y_{n-1})$ “.
p. 593: On the second line, “let $A$ be any other measurable set” should be “let $A$ be an arbitrary set (not necessarily measurable)”.
p. 625, Remark 19.5.2: x and y should be swapped in “equals 1 when $x>0$ and y=0, equals -1 when $x<0$ and y=0, and equals zero otherwise”.

Caution: the page numbering is not consistent across editions.

Thanks to Biswaranjan Behera, Carlos, EO, Florian, Bart Kleijngeld, Eric Koelink, Wang Kunyang, Jason M., Manoranjan Majji, Geoff Mess, Cristina Pereyra, Kent Van Vels, Haokun Xu, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

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87 comments

Comments feed for this article

23 November, 2007 at 10:37 am

Anonymous

Dear Terry,

The AMS url in your post links to another book rather than “Analysis II.” Additionally, I was trying to order the book from Amazon, and was only able to order “Analysis I” through Amazon directly, but couldn’t do the same for “Analysis II”, even though a page exists for the second volume. Any plans on it being sold *directly* by Amazon?

Thanks!

23 November, 2007 at 12:02 pm

Terence Tao

Dear Anonymous,

Thanks for the bad link report. I just checked Amazon and the second volume is currently available (well, there are two copies available, at least):

http://www.amazon.com/gp/product/8185931631/ref=pd_cp_b_0?pf_rd_p=317711001&pf_rd_s=center-41&pf_rd_t=201&pf_rd_i=8185931623&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0ES3B9XEHE3S5E7CKHCS

But it is a rather low volume item, so availability is likely to continue to be spotty.

20 February, 2009 at 4:30 am

carlos

Dear Professor Tao,

First of all I would like you to congratulate you for the two volumes of the book. I had been looking for a reference to revise calculus from scratch and they have proven invaluable.

I would like to point out a few other minor errata I noticed:

p. 471 at the end of the proof |P(y)-g(y)| should be |P(y)-f(y)|
p.477 ex. 15.1.1 corollary 14.6.2 seems a better hint for (e)
p.482 ex.15.2.5 at the end of the identity, the last exponent should be m and not n

I also have a small question concerning Ex. 15.1.1 in page 477. To prove (c) the hint is to use the Weierstrass M-test. However, in the case of a function with an infinite convergence radius (such as for example sin(x) ), the sup norm of each of the f_n is infinite and consequently their infinite sum is not convergent, so I don’t see how the test can be applied. I am probably just making some basic mistake and I would really appreciate it if someone could point it out…

20 February, 2009 at 9:09 am

Terence Tao

Dear Carlos,

Thanks for the corrections! For Ex. 15.1.1(c), one is only asking for uniform convergence on a compact interval [a-r,a+r] (r can be arbitrarily large, but is finite), and in this case the summands are absolultely convergent.

22 March, 2009 at 10:37 am

Florian

Dear Professor Tao,

firstly, I can only agree with Carlos and I congratulate you to your book as well. I especially like the structure of it because you emphasize the importance of the basic elements of analysis (which aren’t obvious at all…).

I have a question concerning your example on page 625. Is the given example function f(x,y) really not (absolutley) integrable over y in x=0?
It would make sense to me, if it wasn’t integrable over x in y=0. Or am I wrong?

Best regards,
Florian

24 March, 2009 at 2:05 pm

Terence Tao

Oops, you’re right, it should be switched. Thanks!

22 April, 2009 at 10:17 pm

Mark

Professor Tau,
I am attempting exercise 12.5.14 in Analysis II and would like to know if you could elaborate on the hint given.

Any help would be greatly appreciated.

Thank you.
Mark

23 April, 2009 at 12:16 pm

Terence Tao

Dear Mark,

By compactness, the sequence $x^{(n)}$ in the hint has a subsequence which converges to a limit $x$ in E. Now compute what the distance of x to $x_0$ is.

23 April, 2009 at 5:51 pm

Mark

Dear Professor Tau,

Thank you for taking the time to assist me.
Your help has been greatly appreciated.

Mark

30 April, 2009 at 9:15 am

Jorge

Dear Professor Tao,

I’m trying to crack Ex. 14.8.2. in your Analysis II but I’m stuck on the part c), namely on the estimation|c*(1 – x^2)^N| <= epsilon which must hold for delta <= |x| <= 1 (this is precisely the condition from part c) in the definition of the approximation to the identity, I mean…). How can I now use part b) of the same Ex. here? Or this is not the case?

Thank you.
Jorge

30 April, 2009 at 10:06 am

Terence Tao

Dear Jorge,

Part (b) ensures that the normalising constant c only grows polynomially in N (or more precisely, is bounded by $C \sqrt{N}$ for some C). On the other hand, once x is bounded away from zero, $(1-x^2)^N$ decays exponentially in N (uniformly in x). As exponential decay always beats polynomial growth, the integral of $c (1-x^2)^N$ away from zero will go to zero.

30 April, 2009 at 10:36 pm

Jorge

So that f_N(x) := c(1-x^2)^N goes to zero function uniformly as N tends to infinity?

1 May, 2009 at 8:54 am

Terence Tao

On any set of the form $\{x: \varepsilon \leq |x| \leq 1 \}$ , yes.

1 May, 2009 at 9:15 am

Jorge

Thank you very much for your time. It is really great thing to be able to consult with you online.

Jorge

23 May, 2009 at 6:31 am

HASSAN JOLANY

HELLO I AM STUDENT UNIVERSITY OF TEHRAN I STUDY ABOUT THE POINCARE CONJECTURE FOR MY THESIS CAN I ASK YOU SOME OF MY PROBLEMS
BEST REGARD FOR YOU

1 June, 2009 at 10:09 pm

Joe

Dear Professer Tao,
I am attempting exercise 14.2.1 (a) from Analysis II. What do I need to show? For the “only if” direction I think I need to show that if f is continuous and the sequence goes to zero then I have pointwise convergence (is this right?), but I am not sure about the “if” direction. Can you give me any help? Thank you,

- Joe

2 June, 2009 at 8:42 am

Mark

Dear Prof Tao,

Is there any way I could get or browse the answers for all the exercises in Analysis 2?

2 June, 2009 at 9:43 am

Terence Tao

Dear Joe,

Yes, this is the “only if” part. The “if part” asserts that if f is a function such that $f_{a_n}$ converges pointwise to f for every sequence $a_n$ converging to zero, then f is necessarily continuous.

Dear Mark: I do not have an answer key for the exercises. Some of the exercises are based on my classes at

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/

which have a partial solution set, but this is far from complete (also, some of the wording of the exercises changed when converted to book form).

2 June, 2009 at 5:05 pm

mark

Thank you prof. Another thing is there any solutions for all the homeworks of your Math131BH?

2 June, 2009 at 7:28 pm

Ali Nesin

Hi prof,

I do come across this q and really need your advise.
Suppose (X,d1), (X,d2) are two metric spaces over the same set X, and c1, c2 are strictly positive.
How do we show that (X,d), where d = c1d1 + c2d2 is also metric space?

2 June, 2009 at 8:44 pm

timur

Dear Ali Nesin,

Try going over the metric space axioms d has to satisfy. Which axiom(s) are you having trouble with?

3 June, 2009 at 1:02 am

Ali Nesin

I do have trouble actually in understanding the q. I think it is the real line, R I’m dealing with. Am I right?

3 June, 2009 at 1:18 pm

timur

Dear Ali Nesin,

I think you should ask your question on a forum like the following (I hope prof. Tao’s does not mind):

http://www.sciforums.com/

http://www.mathlinks.ro/Forum/

You even might find your question already answered.

5 June, 2009 at 6:31 pm

Mark

Mark

Professor Tau,
I am attempting exercise 14.8.8 in Analysis II and would like to know if you could elaborate on the hint given.

Any help would be greatly appreciated.

Thank you.
Mark

6 June, 2009 at 2:12 pm

Anonymous

In the proof of Abel’s Theorem on page 487 is the limit superior suppose to be as y approaches 1 instead of as n approaches infinity?

6 June, 2009 at 8:12 pm

Mark Werner

This is Mark Werner.

I am the original ‘Mark’ in this blog who asked about question 12.5.14. I am not the most recent ‘Mark’ asking about question 14.8.8 which has been set as a take home exam question with no consultation allowed. I believe someone in the class is pretending to be me, since the question was asked in exactly the same way as my original one, including the misspelling your name. I am posting this just to ensure that my name is clear.

Thank you.

- Mark

7 June, 2009 at 10:29 am

student

Dear Prof Tao,

is it possible to find a 1-1 map from [0,1] to (0,1)?

thanks

7 June, 2009 at 10:44 am

timur

Dear student,

There is a general theorem in set theory (called Cantor-Bernstein theorem) that says that for two sets A and B if there are injective maps A -> B and B -> A then there exists a bijective map A -> B. I believe Cantor proved it using AC and later Bernstein gave a constructive proof without AC.

7 June, 2009 at 2:39 pm

student

Thank you Timur,

actually I want to apply that theorem, there is an obvious 1-1 map from (0,1) to [0,1] but I could not find a 1-1 map from [0,1] to (0,1).

if anyone knows, please let me know

thanks

7 June, 2009 at 3:07 pm

timur

Sorry, I misunderstood your question; you can scale and translate [0,1] to, say, [z,1-z] for some z>0 to get an injective map [0,1] -> (0,1).

7 June, 2009 at 3:15 pm

student

good idea thanks Dear Timur

7 June, 2009 at 7:41 pm

geometry beginner

can anyone suggest me a nice introductory book on differential geometry and algebraic topology? my background is just point set topology.

if you know nice introductory survey articles as well, please let me know.

thanks

18 July, 2009 at 5:02 pm

Jason M.

Hello — I had a couple of very small observations:

On page 546, in Definition 17.3.7 (Partial derivative), $f$ is a function defined on a subset of $R^n$, and you are defining the partial derivative with respect to $x_j$, but you index $j$ from 1 to $m$ instead of from 1 to $n$.

Later in that same definition, you write $\lim_{t\to 0;t\neq 0;x_0+tv\in E}$, when I think it ought to be $x_0+te_j$ instead of $x_0+tv$.

Thanks! –Jason

18 July, 2009 at 5:49 pm

Jason M.

Hello again — I have noticed one more thing:

On page 548, in the proof of Theorem 17.3.8, when you are defining the linear transformation L, you index j from 1 to m instead of 1 to n.

These are pretty small details, but I think they could cause confusion for someone unfamiliar with the material!

Thanks! –Jason

[Thanks for the corrections - T.]

20 March, 2010 at 10:31 am

Matt

Dear Professor Tao:

On page 520 of Analysis II, it states that the Fourier Transform is a map from functions in C(R/Z;R). Should the codomain of the functions be the complex numbers? It seems inconsistent with the rest of the surrounding definitions and theorems inasmuch as it only applies to functions with a real codomain. Thank you very much.

Matt

24 June, 2010 at 1:31 pm

Guilherme

Dear Prof. Tao,

I really liked your book. The contents of both volumes are very very good.
I was thrilled. I was thinking omg, this will be my favourite analysis book.

But I saw that you place the proof of some theorems and lemmas as exercises. I really have huge difficulties to learn from books which use this style.

When this happens, it seems that suddenly the author of the book “abandoned me”. And then I get lost. Of course, it a good habit of doing the proofs by myself, but, in the end, I am not sure my proof is good/rigorous or not.

But, the organization and the writing style are amazing !

Best,
Guilherme

24 June, 2010 at 1:41 pm

Terence Tao

This is very much an intentional choice on my part (see page xiii of the preface). There are a lot of important subtleties in real analysis (e.g. the distinction between pointwise convergence and uniform convergence, to give just one example), and the only way to really appreciate this is to chew through some of the foundational results regarding these subtleties. There are some parts of mathematics where it is sufficient to just know the main results and their proofs, as opposed to deriving them with one’s own mental resources, but in my view analysis is not one of these subjects.

24 June, 2010 at 2:09 pm

Guilherme

Dear Prof. Tao,

It is very nice to have this feedback about learning methods from you. And it is indeed true that I learn better when I put some effort in trying to do the proof on my own before seeing the proof from the actual book.

When this happens, I appreciate much more the proof and understand better the overall(big picture) strategy.

On the other hand, what happens if the student doesnt know how to make the proof, or if the student doesnt know if the his proof is wrong ? Don’t you think that the student will miss an important part of the material ?

ps: I am not making this questions with the intention to criticize. Mainly, I am making this questions in order to improve my learning methods too.
I always get frustrated when I realized that the author puts the proofs in the exercises. Maybe it is time for a change. :)

Best,
Guilherme

24 June, 2010 at 2:30 pm

Terence Tao

A student that is unable to make proofs, or to verify whether a given proof is correct, will have difficulty with a subject as theory-intensive as real analysis regardless of whether the textbook contains proofs of its theorems or not, particularly when one reaches the point where one needs a result not already contained in the textbook.

In the case of my own textbook, in most cases when a proof of a theorem is set as an exercise, a proof of a related or similar result is provided in the text as a model; the ability to adapt an argument from one context to another is another important skill in analysis that is difficult to pick up by any means other than direct practice.

24 June, 2010 at 6:24 pm

Tom

Dear Terry,

Can one use matrices and linear algebra to explain both pointwise and uniform convergence?

9 August, 2010 at 5:17 am

Bart Kleijngeld

It’s only a minor correction, but they all count, right? On page 536, example 17.1.7 you write $x \in \mathbb{R}$ , while it should be $c \in \mathbb{R}$ .

Thanks for your great book,
Bart

[Corrected, thanks - T.]

7 September, 2010 at 7:30 pm

Student

Dear Prof. Tao, I am having trouble with 13.2.10 in analysis 2. I have to show that
dx((x,y), (x0, y0)) is greater than dx((x,y0), (x0, y0)) (holding one variable constant). It seems rather obvious, but I cannot seem to get it.

7 September, 2010 at 7:36 pm

Terence Tao

That’s not quite the right approach. I recommend that you write down in full (using epsilons and deltas) what it means for the two-variable function $(x,y) \mapsto f(x,y)$ to be continuous at $(x_0,y_0)$ , and what it means for the function $x \mapsto f(x,y_0)$ to be continuous at $x_0$ , taking particular care to get all the quantifiers (“for all”, “there exists”) correct and in the right order. Once you have both definitions down side by side, it should become clear how to proceed.

14 September, 2010 at 3:11 pm

Juan

Hello Dr. Tao,

I am having trouble with 13.4.6 in your book, Analysis II. I have been trying to do a proof by contradiction, assuming that the union of an arbitrary number of connected sets is in fact disconnected, but I am stuck.

Here is what I have:

Call the arbitrary union of the connected sets, E.
Assume that there exist open, disjoint, nonempty sets A, B, such that A union B = E.

Then A = complement of B.

Ea (the alpha’th set in the collection of connected sets). Assume Ea is not contained in A nor B (rather it is “split up” among the two sets). Thus there exists a subset of A, call it D, and a subset of B, call it E, s.t. D union E = Ea. But Ea is connected, so D and E are either nondisjoint, or one of the sets is not open. That is where I am stuck. What do I do from there? I assume that I want to show that for all alpha, Ea must be contained in either A or B, so then we get a contradiction, but I do not know.

14 September, 2010 at 7:56 pm

Terence Tao

If A is open in E, then $D := A \cap E_\alpha$ will be open in $E_\alpha$ .

It may help to isolate a common point $x_0$ of the intersection $\bigcap_\alpha E_\alpha$ , which is non-empty by hypothesis. Note that $x_0$ must belong to exactly one of A or B.

4 November, 2010 at 9:04 am

Kent Van Vels

Hello Dr. Tao,

I am going through your analysis book. I may have found a typo in the second edition. In exercise 16.5.4, should the formula be

$\widehat{f'(n)} = i n \widehat{f}(n)$ be changed to

$\widehat{f'(n)} = 2\pi i n \widehat{f}$ ?

Thanks,

Kent

[Added, thanks - T.]

4 November, 2010 at 9:06 am

Kent Van Vels

I mean, of course,

$\widehat{f'}(n) = 2\pi i n \widehat{f}(n)$ .

13 November, 2010 at 11:10 am

Kent Van Vels

I have a question the second edition of Analysis II.
For the proof of Proposition 14.1.5 (exercise 14.1.2)

Can you explain where my reasoning goes wrong in showing that the third statement is logically distinct from the first statement?

If we consider the function $f:\mathbb{R}\to \mathbb{R}$ defined as $f(x) = 0$ for $x\not =0$ , and $f(x) = 1$ for $x=0$ . Then
$\lim_{x\to 0; x\in \mathbb{R}} f(x) = 0$ .
Now, let $V=(-1/2,1/2)$ , an open set which contains $0$ . Now, every open set that contains $0$ will contain an interval of the form $(-a,b)$ where $a,b>0$ . But $f((-a,b) \cap \mathbb{R}) = \{0,1\} \not \subset (-1/2,1/2)= V$

Do we need to assume that $f$ is continuous or that $x_{0}$ is a limit point (instead of an adherent point)?

Thanks,

Kent Van Vels

13 November, 2010 at 12:14 pm

Terence Tao

For your specific function, the limit $\lim_{x \to 0; x \in {\bf R}} f(x)$ does not exist, because you are not excluding 0 from the range of possible values of x. (This notational convention may be a bit different from what is seen in some other texts; see Remark 14.1.2.)

13 November, 2010 at 1:11 pm

Kent Van Vels

Thank you. That clears up my confusion.

23 November, 2010 at 2:55 pm

Anonymous

Prof. Tao,

After studying the Chapter 17 Several variable differential calculus in Analysis vol.2 , I am curious about whether there are high order derivatives for the function $f: R^n\rightarrow R^m$ . But a further consideration seems to tell me that the high order derivative is not the transformation as $L_{Df}$ any more. It seems that one has to define the “second derivative” for the transformation $L_{Df}:R^n\rightarrow R^{m\times n}$ . But it seems strange.

Could you explain about this? If they do exist, what’s the topic there in the larger picture? Or is this concept meaningless?

23 November, 2010 at 8:12 pm

Terence Tao

These higher order derivatives are best described either as multilinear maps or as tensors. For instance, the second derivative is a bilinear map that is essentially the Hessian. The third and higher derivatives can also be defined, but they are less pleasant to work with, basically because multilinear algebra is significantly less pleasant than linear algebra in many ways. (Also, when working on a curved domain, the underlying curvature of the space begins to assert its influence much more strenuously on the third and higher derivatives than on the first and second ones, leading to further technical complications.)

30 November, 2010 at 11:09 am

Kent Van Vels

In Lemma 18.4.8 (the proof of countable additivity of Lebesgue measure), The third sentence that begins “Let $A$ be any other measurable set…”. Is it ok to only consider measurable sets $A$ to show that $E$ measurable?

[Oops, $A$ should be an arbitrary set rather than a measurable one. I've added an appropriate erratum - T.]

1 December, 2010 at 2:50 pm

Kent Van Vels

Can you explain where my reasoning goes wrong in showing a
counterexample to Exercise 13.5.5?

Let $X$ be the set $\{0,1\}\subset \mathbb{N}$ , with the $0\prec 1$ . Then $X$ is totally ordered. But for
$x=0$ (an element of $X$ ) we can’t find an $a\in X$
such that $a\prec x$ . This shows
that $X$ isn’t open, and so $(X,\mathcal{F})$ isn’t a
topology.

1 December, 2010 at 4:16 pm

Terence Tao

Oops, one also needs to allow for half-rays and the whole space in addition to intervals. I’ve added an appropriate erratum.

1 December, 2010 at 7:32 pm

Kent Van Vels

Should Exercise 13.5.6 and Exercise 13.5.7 be changed to read that the topologies in question are non-Hausdorff? It seems impossible for two disjoint open sets to exist.

[Corrected, thanks - T.]

2 December, 2010 at 11:00 am

KVV

In Proposition 12.4.12 it says

“Let $(X,d)$ be a metric space…” should this be changed to read

“Let $(X,d)$ be a complete metric space?

Thanks.

2 December, 2010 at 11:17 am

Terence Tao

X needs to be complete for part (b), but this is not needed for part (a).

4 December, 2010 at 11:01 pm

Kent Van Vels

Thanks. That clears up my misunderstanding.

Kent

4 December, 2010 at 12:59 pm

KVV

In exercise 12.5.12 there is a hint that reads in part,

“…useless here since that only applies to Euclidean spaces with the Euclidean metric.” Should that be changed to indicate that the Heine-Borel theorem also applies to Euclidean spaces equipped with the sup-norm and taxi-cab metric?

Thanks,

Kent

[Corrected, thanks - T.]

12 December, 2010 at 10:52 am

KVV

On page 515 in the second edition there might be two small typos.

There is math display that reads, $latex\cdots \sum_{j=\in J}\cdots$, I don’t think the equals sign is required.

Also, in the definition of the outer measure, there is a sum from $j=1$ to $\infty$ . Should this be a sum over $j\in J$ ?

Thanks,

Kent

[Corrected, thanks - T.]

13 December, 2010 at 8:24 am

KVV

Hello,

On page 545, in the statement of Corollary 19.2.11, there is a sequence of functions. Do we have to assume that each function in the sequence is measurable?

Thanks,

Kent

[Corrected, thanks - T.]

29 December, 2010 at 1:07 pm

KVV

I think there is a small typo in the statement of Exercise 15.2.8. For part (e), there is a sum whose summand is $d_m (x-b)^n$ , I think it should probably be $d_m (x-b)^m$ .

Thanks.

[Corrected, thanks - T.]

17 January, 2011 at 3:22 pm

KVV

I think there are two small typos in the statement of exercise 14.1.6 on page 396, (second edition).

First, there is a display $\lim_{y\to y_0; y\in f(E)} g(x) = z_0$ , I think it should probably be $\lim_{y\to y_0;y\in f(E)} g(y) = z_0$ .

Second, there is a display $\lim_{x\in x_0; x\in E} g \circ f(x_0) = z_0$ , I think it should probably be $\lim_{x\to x_0; x\in E} g\circ f(x) = z_0$

Thanks

[Corrected, thanks - T.]

13 March, 2011 at 10:56 pm

Biswa

Dear Prof. Tao,

On Page 478, just before Definition 15.2.4, “…and differentiable on (a-r, a+r) for some $a\in{\mathbb R}$” should be “…and differentiable on (a-r, a+r) for some $r>0$.

Thank you.
Biswaranjan Behera

[Corrected, thanks - T.]

15 March, 2011 at 3:26 am

Biswa

P 485, $s_N$ (third display) and $s_0$ (next line) should be
$S_N$ and $S_0$ . Also, should it be “by convention
$S_0=-D$ in place of “in particular $S_0=-D$ ?

[Corrected, thanks. The way summation has been defined in the text, an empty summation is automatically zero without the need for any explicit further convention. -T.]

30 April, 2011 at 1:52 pm

Anonymous

I am confused by the exercise 14.1.1. Why “the limit $lim_{x\to x_0;x\in E}f(x)$ exists iff the limit $lim_{x\to x_0;x\in E\setminus \{x_0\}}f(x)$ AND is equal to $f(x_0)$ ?” This is true only when $f:E\to Y$ is continuous, isn’t it?

30 April, 2011 at 11:55 pm

Terence Tao

No, continuity is not required for this fact. (Both the hypothesis and the conclusion of the exercise, though, are equivalent to $f$ being continuous at $x_0$ .)

1 May, 2011 at 8:43 pm

Anonymous

Dear Prof. Tao,

I am not sure if this is the right place to put the following question here.

In the very beginning of the book, Chapter 1 of volume I, you introduce what is real analysis and point out that it is the theoretical foundation “which underlies calculus”. However, I didn’t find that you talk about the “multivariable calculus” in you real analysis series courses, from math131AB to math 245ABC.

Actually, you did talk about the “several variable differential calculus” in Chapter 17 in this volume. But why there is no “multivariable integration” here? Is it because it is enough for the theory which you have talked about in your series books(Analysis I,II; An introduction to measure theory;An epsilon of room, Vol I) to build up the “multivariable integration”, or because it is beyond the scope of this course?

Since I’d like to build up the system of knowledge about real analysis myself, I follow the series of your books. But it puzzles me that I am even not able to find the “multivariable integration” in your books.

1 May, 2011 at 9:34 pm

Terence Tao

See Chapter 19.

1 May, 2011 at 9:55 pm

Anonymous

I did not put my question in the right way. I think I am talking about the [multiple integral](http://en.wikipedia.org/wiki/Multiple_integral) and the [vector analysis](http://en.wikipedia.org/wiki/Vector_calculus). In the one dimensional case, you talked about the Riemann integral in Chapter 11 and finally the Lebesgue integration in Chapter 19. In the n-dimensional case, you introduce the Lebesgue integration in ${\bf R}^n$ in Chapter 19. So there is no “Riemann integral” but the Lebesgue integration in the multi-dimensional case? What’s the theoretical counterpart of multiple integral in the vector analysis in your book?

2 May, 2011 at 7:08 am

Terence Tao

See Section 19.5.

Generalising the one-dimensional Riemann integral as set up in Chapter 11 to higher dimensions would be a good exercise for a student who has gone through that chapter. (I explicitly set that exercise, incidentally, as Exercise 1.1.26 of my measure theory book.) However, I feel that this is something that a good student can work out on his or her own, after understanding the core topics covered in texts such as my own (as well as seeing a lower-division treatment to higher-dimensional integration).

4 June, 2011 at 3:36 pm

Anonymous

I find that you didn’t deal with the “Local maxima, local minima” in the multivariate case in these two volumes. Is this intentional? Will you recommend any references which deal with this topic rigorously?

12 June, 2011 at 8:08 am

Anonymous

Errata to the second edition, ‘formula does not parse’

[Corrected, thanks - T.]

12 June, 2011 at 9:23 am

Anonymous

Dear Prof. Tao,
I am confused with the definition of partial derivative(p.546 Definition 17.3.7). Is it a special case of the “directional derivatives”? If it is, why the constrain for $t$ in the definition is $t\to 0;t\neq 0$ instead of $t\to 0;t>0$ which appears in the definition of directional derivative? If it is not, how can you conclude from Lemma 17.3.5 that $\frac{\partial f}{\partial x_j}(x_0)=f'(x_0)e_j$ ?

12 June, 2011 at 10:49 am

Terence Tao

The partial derivative $\frac{\partial f}{\partial x_j}$ is related to the two directional derivatives $D_{e_j} f$ and $D_{-e_j} f$ in that whenever $D_{e_j} f(x)$ and $-D_{-e_j} f(x)$ exist and are equal to each other, then the partial derivative $\frac{\partial f}{\partial x_j}(x)$ also exists and is equal to these two quantities, and conversely (this follows from Proposition 9.5.3).

12 June, 2011 at 7:15 pm

Anonymous

Dear Prof. Tao,

For the Exercise 17.3.3, by definition, $D_vf(0.0)=\frac{v_1^3}{v_1^2+v_2^2}$ where $v=(v_1,v_2)$ . and ${\partial}{\partial x}f(0,0)=1$ , ${\partial}{\partial y}f(0,0)=0$ . Then by your remark in p.547, $D_vf(0,0)=\sum_jv_j\frac{\partial f}{\partial x_j}(0,0)$ , $f$ is not differentiable at $(0,0)$ . Is the proof supposed to be like this? Or do you have any hint for this?

[This will work, yes - T.]

12 June, 2011 at 7:19 pm

Anonymous

Dear Prof. Tao,
I didn’t see any connection between Exercise 17.3.1 and Exercise 17.1.3. (You said that “this will be similar to Exercise 17.1.3″)Typo?

[Sorry, this should be Exercise 17.2.1. -T.]

15 June, 2011 at 9:20 am

Chris

Dear Professor Tao

I had another confusion that require clarification.

1) In your above errata, you mention, Pg 478: Just before definition, ………. should be ………..

My comment: The actual page is 427 but I can’t find the one that need to be replace. Can you please justify this?

2) As to your statement on Pg 485, ………….

My Comment: The correction can be made by me because the page number is wrong and I can’t identify where is that part in the text that need to be replaced.

3) Pg 548 you mention that m should be n. Can you clarify further as to what should be replaced in the text.

4) Pg 593 & Pg 625 .<— For this errata, there is no such pages in your second edition book. Can you justify which should be corrected in the second edition as I can't locate what need to be corrected.

[All errata for the first edition should already be corrected for the second edition - T.]

16 June, 2011 at 7:12 pm

Anonymous

In the section 17.8: The implicit function theorem, p.568, the notation is a little confusing for me. In the first paragraph, $f:{\bf R}^n\to{\bf R}$ gives rise to a graph $\{(x,f(x)):x\in{\bf R}^n\}$ in ${\bf R}^{n+1}$ . And in the second paragraph, $f$ is used in “a hypersurface of the form $\{x\in{\bf R}^n:f(x)=0\}$ , where $f:{\bf R}^n\to {\bf R}$ is some function”. Are these two $f$ the same?? I guess the $f$ in the theorem 17.8.1 is the same as that in the second paragraph, while $g$ is the one in the first paragraph? Am I right?

[Yes, that is right; I will change the first f to a g to reduce confusion. -T.]

9 August, 2011 at 2:56 am

Anonymous

p. 563, line 10: “We remark that …if $f(x_0$ is” should be “We remark that …if $f'(x_0$ is”.

Biswa

[Added, thanks - T.]

9 August, 2011 at 9:48 pm

Anonymous

p. 560, Lemma 17.6.6, Can one apply the contraction mapping theorem to F as it is defined on B(0, r)?

[Oops; I've adjusted the argument there accordingly. -T.]

9 August, 2011 at 9:59 pm

Anonymous

p. 567, line -2, (0, 1) should be (-1, 0).

[Added, thanks - T.]

12 August, 2011 at 9:21 pm

Anonymous

p.563, line 17, ‘ $f^{-1}(y)={\tilde f}^{-1}(y+f(x_0))$ ‘ should be ‘ $f^{-1}(y)={\tilde f}^{-1}(y-f(x_0))$ ‘..

p.570, second line of the last paragraph, ‘U is open and contains $(y_1, \dots,y_{n-1},0)$ ‘ should be ‘U is open and contains $(y_1, \dots,y_{n-1})$ ‘.

Biswa

8 February, 2012 at 9:02 pm

Anonymous

Can anyone show me if this integral $\int_{\pi}^{\3pi/2}\frac{1}{x^2 \sin^{3/2}(x)}dx$ is finite or not?

Thanks

5 May, 2012 at 5:50 am

ugroh

Terence, two comments on Analysis 2:

p.25 Theorem 15.1.6. (d): Should not the formula for $f'$ starts with
$n=1$ instead of $n=0$ ? This would be more consistent with Prop. 15.2.6. and the formula there.

p.427, 3rd line from the top: … differentiable on .. for some $r \in R$ instead of $a \in R$ ?

Ulrich

[Corrections added, although I will have to recheck the page numbers on the errata list the next time I have access to a physical copy of the book, as some of the numbers seem to be incorrect.]

6 May, 2012 at 9:58 pm

ugroh

Sorry, it should be p. 425 not p. 25
Ulrich

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