Metric space question and solution

Reference: I had blogged this example earlier, but I myself could not fill in the missing gaps at that time. I am trying again with the help of MathWorld Wolfram and of course, the classic, Introductory Real Analysis by Kolmogorov and Fomin, from which it is picked up for my study.

Consider the set C_{[a,b]} of all continuous functions defined on the closed interval [a,b]. Let the distance function (or metric) be defined by the formula:

\rho(x,y) = (\int_{a}^{b}[x(t)-y(t)]^{2}dt)^{1/2} ——– relation I

The resulting metric space will be denoted by C_{[a,b]}^{2}.

The first two properties of a metric space are clearly satisfied by the above function. We need to check for the triangle inequality:

Now I satisfies the triangle inequality because of the following Schwarz’s inequality:

(\int_{a}^{b}x(t)y(t)dt)^{2} \leq \int_{a}^{b}x^{2}(t)dt \int_{a}^{b}y(t)dt —— relation II

In order to get to the above relation II, we need to prove the following:

Prove: (\int_{a}^{b} x(t)y(t)dt)^{2} = \int_{a}^{b}x^{2}(t)dt \int_{a}^{b} y^{2}(t)dt - \frac{1}{2}\int_{a}^{b} \int_{a}^{b}[x(s)y(t)-x(t)y(s)]^{2}dsdt.

From the above, we can deduce Schwarz’s inequality (relation II here in this blog article).

(My own attempts failed to crack it so I had to look at the internet for help. Fortunately, MathWorld Wolfram has given a crisp clear proof…but some parts of the proof are still out of my reach…nevertheless, I am reproducing the proof here for the sake of completeness of my notes…for whatever understanding I can derive at this stage from the proof…):


Weisstein, Eric W. “Schwarz’s Inequality.” From MathWorld–A Wolfram Web Resource.

Schwarz’s Inequality:

Let \Psi_{1}(x), \Psi_{2}(x) be any two real integrable functions in [a,b], then Schwarz’s inequality is given by :

|< \Psi_{1}, \Psi_{2}>|^{2} \leq < \Psi_{1}|\Psi_{2}> <\Psi_{2}|Psi_{1}>

Written out explicity,

|\int_{a}^{b} \Psi_{1}(x), \Psi_{2}(x)|^{2} \leq \int_{a}^{b}[\Psi_{1}(x)]^{2}dx \int_{a}^{b}[\Psi_{2}(x)]^{2}dx

with equality if and only if \Psi_{1}(x) = \alpha \Psi_{2}(x) with \alpha a constant. Schwarz’s inequality is sometimes also called the Cauchy-Schwarz inequality or Buniakowsky’s inequality.

To derive the inequality, let \Psi(x) be a complex function and \lambda a complex constant such that

\Psi(x) \equiv f(x) + \lambda g(x) for some f and g


\int \overline{\Psi} \Psi dx \geq 0, where \overline{z} is the complex conjugate.

\int \overline{\Psi}\Psi dx = \int \overline{f}f dx + \lambda \int \overline{f} g dx + \overline\lambda \int \overline{g} f dx + \lambda \overline{\lambda} \int \overline{g} g dx

with equality when \Psi(x) = 0

Writing this, in compact notation:

<\overline{f},f> + \lambda <\overline{f},g> + \overline{\lambda} <\overline{g},f> + \lambda \overline{\lambda} <\overline{g},g> \geq 0….relation A

Now, define \lambda \equiv - \frac{<\overline{g}, f>}{<\overline{g},g>}….relation B

and \overline{\lambda} = - \frac{<g, \overline{f}>}{<\overline{g}, g>}…relation C

Multiply A by <\overline{g},g> and then plug in B and C to obtain:

<\overline{f}, f><\overline{g}, g> - <\overline{f},g><\overline{g},f> - <\overline{g},f><g, \overline{f}> +<\overline{g}, f><g, \overline{f}> \geq 0

which simplifies to

<\overline{g},f><\overline{f},g> \leq <\overline{f},f><\overline{g},g>


|<f,g>|^{2} \leq <f,f><g,g>. Bessel’s inequality follows from Schwarz’s inequality. QED.


Nalin Pithwa

BIG Math Network Blog:

I hope to start writing this blog regularly, if not frequently, but, firstly, I hope the following blog helps many of my readers too:

BIG Math Network: Connecting Mathematical Scientists in Business, Industry, Government and Academia

(I just came across today from an AMS Blog.)

(Thanks to Evelyn J. Lamb)

— shared here by Nalin Pithwa.

Misteaks of mathematicians


(As remembered by Prof Walter Rudin in his autobiography, “The Way I Remember it”.)

Mathematicians are human. Humans make mistakes. Therefore…

This is no cause for alarm. I have no figures to back this up, but compared to the flood of published papers, the number of serious errors in the literature must be tiny. Most are probably caught by referees. And, if there is a serious error in a paper that is important enough to be studied by a significant number of interested mathematicians, that error will be discovered. Even better, the one who made it won’t be able to argue his way out of it.

There is an amusing article by Geoffrey K. Pullum in Natural Language and Linguistic Theory 5, 1987, 303-309, which compares this social aspect of mathematics with what happens in linguistics. It describes the story of Rourke’s claim to have proved the Poincare conjecture. (The article was sent to me by Catherine, my linguist daughter.)

My first encounter with this sort of thing started with the letter on the following page (typed a bit later after two paragraphs):

Needless to say, I was totally amazed. Here was Dieudonne, a world class mathematician and one of the founders of Bourbaki, not telling me, a young upstart, “you are wrong, because here is what I proved a few years ago” but asking me, instead to tell him what he had done wrong! Actually, it took me a while to find the error, and if I had not proved earlier (in J. Math. Mech. 7, 1958, 103-116) that convolution-factorization is always possible in L^{1} I would have accepted his conclusion with no hesitation, not because he was famous, but because his argument was simple, and, perfectly correct, as far as it went.

He proved, correctly, that every convolution of non-negative functions coincides almost everywhere with one that is lower semicontinuous. But (and this is what he ignored) that function may have +\infty among its values.


Evanston, Illinois.

58, Rue de Verneuil, Paris 7e (France).

Paris, December 17

 Dear Professor Rudin:

In the last issue of the Bulletin AMS, I see that you announce in abstract 7311, p.382, that in the algebra L^{1}(R^{n}), any element is the convolution of two elements of that algebra. I am rather amazed at the statement, for a few years ago I had made a simple remark which seemed to me to disprove your theorem (Compositio Mathematica, 12 (1954) p.17, footnote 3). I reproduce the proof for your convenience.

Suppose f, g are in L^{1} and greater than or equal to zero, and for each n consider the usual “truncated” functions f_{n}=\inf(f,n) and g_{n}=\inf(g,n); f(resp. g) is the limit of the increasing sequence (f_{n}) (resp. g_{n}), hence, by the usual Lebesgue convergence theorem, h=f*g is a.e. the limit of

h_{n}=f_{n}*g_{n}, which is obviously an increasing sequence. Moreover, f_{n} and g_{n} are both in L^{2}, hence it is well known that h_{n} can be taken continuous and bounded. It follows that h is a.e. equal to a Baire function of the first class. However, it is well-known that there are integrable functions which do not have that property, and therefore they cannot be convolutions.

I am unable to find any flaw in that argument, and if you can do so, I would very much appreciate if you can tell me where I am wrong.

Sincerely yours

J. Dieudonne

His argument proves that there are non-negative functions h in

L^{1} that are not representable as h=f*g with f \geq 0 and g \geq 0. (I had also observed this). In the general real-valued case, if h=f*g, each of f and g is a difference of two non negative functions, so that f*g breaks into four convolutions of the type considered by Dieudonne. Of these, two are greater than or equal to zero, two are less than or equal to zero, and one may therefore run into the problem of subtracting \infty from \infty (which is at least as much as of a no-no as is dividing 0 by 0). Hence, one can no longer conclude that h coincides almost everywhere with a function of Baire class one, that is, with a real-valued function which is everywhere the pointwise limit of a sequence of continuous ones.

Dieudonne had fallen into the “without loss of generality” trap by restricting himself to f \geq 0 and g \geq 0, and tacitly assuming that the general case would follow.

Here is his  reply to my explanation.

Paris, January 12, 1952.

Dear Professor Rudin:

Thank you for pointing out my error; as, it is of a very common type, I suppose I should have been able to detect it myself, but you know how hard it is to see one’s own mistakes, when you have once become convinced that some result must be true!!

Your proof is very ingenious; I hope you will be able to generalize that result to arbitrary locally compact abelian groups, but I suppose this would require a somewhat different type of proof.

With my congratulations for your nice result and my best thanks, I am

Sincerely yours

J. Dieudonne

The factorization theorem was indeed extended, even further than he had hoped. When Paul Cohen saw my rather complicated proof about L^{1}(\Re^{n}) (the case n=1 was much easier) he said: “Aha, approximate identities” and quickly produced a very general factorization theorem in Banach algebras with approximate left identities (Duke Math. J. 26, 1959, 199-205). Ed Hewitt (Math. Scand. 15, 1964, 147-155) extended Cohen’s proof so as to include convolution operators on L^{p} (1 \leq p < \infty). Every h in L^{p} is f*g with f in L^{1}, g in L^{p}.

In my next story, I was the one who goofed, but it ended well. This concerned the open unit ball B in the n-dimensional complex space C^{n}. A one-to-one holomorphic map from B onto B will be called an automorphism of B. (When n=1, B is the unit disc in C, and its automorphisms are the familiar Moebius transformations that send z to e^{i\theta}\frac{(z-\alpha)}{(1-\overline{\alpha} z)} ). The automorphisms of B are also explicitly known for all n.

Can a space X of complex valued functions on B(or on the boundary of B)  Moebius-invariant (or \mathcal{M} invariant) spaces of certain types (Duke Math. J. 43, 10976, 841-861) but the following was not answered:

Which closed subalgebras of C(B) are \mathcal{M}- invariant?

Here, C(B) is the algebra of all complex valued continuous (possibly unbounded) functions on B, with the topology of uniform convergence on compact sets, and with pointwise addition and multiplication.

The five obvious possibilities are: \{ 0 \}, the constants, the holomorphic functions on B, those whose complex conjugates are holomorphic and C(B) itself. The answer (Ann. Inst. Fourier 23, 1983, 19-41) is:

Theorem. There are no others.

I believe that this is the most difficult theorem that I ever proved. It was new even for n=1. I started with a proof of the one-dimensional case, and then used that to derive the same conclusion in n dimensions. Fairly soon after I submitted the paper, Malgrange, who was an editor of Annales Fourier, wrote that the referee did not understand how I passed from 1 to n. When I looked at it, I couldn’t understand it either! What I had written simply made no sense, and there seemed to be no way to repair it. I had to go back and instead of first dealing with the one-variable case I had to do the whole thing in n variables from the start. Fortunately, it worked. But it took a whole summer.

When I sent the corrected (much longer) version to Malgrange, I wrote that I should like to  thank the referee in the corrected paper, but only if I could mention his or her name. I saw no virtue in anonymous thanks. The referee agreed to this; it turned out to be my friend Jean-Pierre Rosay.

A few years later, we (i.e., the Madison Mathematics Department) wanted to invite Rosay for a whole year’s visit. I had a so-called Vilas Professorship which provided research funds for worthy projects. So I sent a request to the appropriate committee; to strengthen the case, I mentioned that not only I but several of my colleagues (Ahern, Forelli, Nagel, Wainger among them) would find him very stimulating. My letter was returned by no other than Irv Shain, the Chancellor, saying that Vilas Professorships were only for the benefit of those who  had one and for no one else’s, and that I should write a different letter, explaining how Rosay’s presence would benefit me. I did that, and as a clincher enclosed a reprint of the paper in which I thanked him. That did it.

Soon after he arrived, he and I started  to work on several questions about holomorphic maps. This resulted in a long paper. (Trans. AMS 310, 1988, 47-86), the first of several that we wrote together. Our collaboration went so well that I suggested that we ought to try to  keep him. We succeeded in this, and it could well be that my role in getting him to stay here was one of thei best things I ever did for the Department.

I know of only one totally absurd paper that was published in a respectable journal, namely the one by Nikola Pandeski in Math., Annalen 287, 1990, 185-192. In one variable, the “corona theorem” asserts that the open unit disc U is dense in the maximal ideal space of the Banach algebra of all bounded holomorphic functions in U. Its original proof, by Lennart Carleson (Annals of Math. 76, 1963, 547-559) involved a great deal of difficult “hard” analysis. A much simpler one was later found by Tom Wolff. But the n-variable analogue, which Pandeski claimed to have proved, is still wide open.

This paper appeared during a several complex variables conference in Oberwolfach. I heard that it caused great hilarity because nothing in it makes sense. For example, the proof starts by covering a sphere with a finite disjoint collection of small balls! I have heard several explanations about how this absurdity got into print, none of them convincing. I was quite annoyed about the whole affair because Pandeski attributed several absolutely false assertions to me, and because Granert, the editor who  had (mis)handled this paper refused to publish my protest.


Never mind. To err is human …

Nalin Pithwa

Five things to do as a graduate student in Mathematics

The following are the views of Mr. Mohammed Kaabar posted on AMS Graduate Blog just today:

I would like to share with you my first year experience as a graduate student in mathematics at Washington State University, and I want to give you some suggestions about what you should do as a graduate student in mathematics?. In Spring 2015, I started my first semester as a Ph.D student in Applied Mathematics, and during that semester, I wrote two math textbooks in differential equations and linear algebra, and I also gave three seminar’s talks in Applied Mathematics, as well as, I participated as invited technical program committee (TPC) member and invited reviewer for many international conferences and journals in applied math, physics, electrical engineering, and computer engineering. Most of these conferences published their accepted papers in major trade peer-reviewed publishing companies such as Springer and IEEE Xplore. Therefore, the following is a list of five different things that I highly recommend you to do as a graduate student in mathematics:

  • Join professional organizations in mathematics and other related fields: When I started my graduate studies in mathematics, I joined the Society for Industrial and Applied Mathematics (SIAM). It is easy to become a student member in SIAM because they offer a free membership for graduate students in some universities. My university was one of them, so I got a free membership. There are several math associations and societies such as American Mathematical Society (AMS) and Mathematical Association of America (MAA) that can provide you with good discounts on the prices of their student memberships.
  • Create a professional website: If you are a newly admitted graduate student in mathematics, I recommend you to create a professional website that includes your research interests, curriculum vitae, work experience, and courses you are currently teach. The advantage of having your own professional website is that many people will contact you by your website email to invite you as technical program committee (TPC) member, reviewer, editor, math team member for conferences and journals in mathematics and applied sciences. If you are going to teach a class, it is a good idea to add a section in your professional website that contains your lecture notes, solutions to your class assignments and quizzes, and study guides for exams.
  • Teach a course you like to teach: If you have been offered a teaching assistantship position at your department, I believe that most universities give you the option to request the courses you like to teach. Therefore, I recommend you to choose courses that interest you more than others because if you like the course you teach, your students will more likely appreciate the way you teach.
  • Participate in research groups: If you are a new graduate student in your department, I recommend you to contact your department chair, coordinator, advisor, and graduate studies chair to ask them about any available research groups to join them as a member so you can participate in the group’s research publications and seminar’s talks.
  • Participate in extra-curricular and academic related activities: When you start you graduate studies in mathematics, you will have a stress of work and study load. So, what you should do to relief this stress?. The answer is simple; many universities and colleges have student’s associations and clubs such as graduate student association and peer leadership program. For example, when I was student at Washington State University (WSU) and American University of Sharjah (AUS), I was an active member in peer leadership program, and I had also taken part taken part in competitions such as The International Electronics Synopsys Competition.

In conclusion, from the fifth point I mentioned above, I would like to focus on one case which I consider a great achievement in my work as a peer leader. One day, while I was walking along the corridors of AUS, I saw a student wondering, knowing neither where to go nor what to do. That student was as perplexed as a person going astray in the desert without being able to decide his direction. I approached him and asked him what he wanted. He told me that it was his first day at AUS and he did not know where and how to start. I assumed him that everything would be alright. Then, that student released a sigh of relief exactly the same feeling of our friend in the desert when a plane out of the blue sky took him out of the mire. That freshmen student was like a ship in a rough sea beaten by high waves, sometimes taking it west and some other times right. Imagine what would that person feel when he suddenly finds someone to lead him to the shores of safety. I helped him throughout the registration process. Since then, that student became one of my best friends. Maybe you are still thinking of our poor friend in the desert? Relax; he was lifted by a helicopter. So my job strengthens relations and builds a highly cooperative community. During my work as peer leader, I oftentimes go around talking to students, familiarizing myself with their problems and offering them the help they may stand in need of. This is just to show you an example of how a successful graduate student can positively impact the lives of other students. Finally, I recommend you to follow at least most of the five things mentioned above to be successful in your career as a graduate student.


If you like it, please send a thank you note to Mr. Mohammed Kaabar on the AMS blog.

More later,

Nalin Pithwa

Chapter 1: Real Variables: examples II

Examples II.

1) Show that no rational number can have its cube equal to 2.

Proof 1.

Proof by contradiction. Let x=p/q. q \neq 0, p, q \in Z. (p, q have no common factors).

Let x^{3}=2. Hence, \frac{p^{3}}{q^{3}}=2. Hence, p^{3}=2q^{3}. Hence, p^{3} is even because we know that even times even is even and even times odd i also even and odd times odd is odd. Hence, p ought to be even. Let p=2m. Then, again q^{3}=4m^{3}. Hence, q^{3} is even. Hence, q is even. But, this means that p and q have a common factor 2 which contradicits our hypothesis. Hence, the proof.. QED.

Proof 2)

Let given rational fraction be \frac{p}{q}, q \neq 0, p, q \in Z.

Let \frac{p}{q}=\frac{m^{3}}{n^{3}}, n \neq 0, m,n \in Z.

Since p and q do not have any common factors, m and n also do not have any common factors.

Case I: p is even, q is odd so clearly, they do not have any common factors.

Case IIL p is odd, q is odd but with no common factors.

Case I: since m and n are without any common factors, and m^{3}, n^{3} are also in its lowest terms, we have p=m^{3}, q=n^{3}.

Case II: similar to case I above.

Proof 3.

A more general proposition, due to Gauss, includes those two above problems as special cases. Consider the following algebraic equation;

x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0.

with integral coefficients,, cannot have a rational root but non integral root.

Proof 3:

For suppose that the equation has a root a/b, where a and b are integers without a common factor, and b is positive. Writing a/b for x, and multiply both the sides of the equation b^{n-1}, e obtain

-\frac{a^{n}}{b}=p_{1}a^{n-1}+p_{2}a^{n-2}b+\ldots +p_{n}b^{n-1},

a fraction in the lowest terms equal to an integer, which is absurd. thus, b=1, and the root is a. It is clear that a must be a divisor of p_{n}

Proof 4.

Show that if p_{n}=1 and neither of

1+p_{1}+p_{2}+p_{3}+\ldots,, 1-p_{1}+p_{2}-p_{3}+\ldots is zero, then the equation cannot have a rational root.

I will put the proof later.

Problem 5.

Find the rational toots, if any of x^{4}-4x^{3}-8x^{2}+13x+10=0.’


The roots can only be integral and so \pm 1, \pm 2, \pm 3, \pm 5 pm 10 are the only possibilities: whether these are roots can be determined by tiral. it is clear that can in this way determine the rational roots of any equation.

More later,

Nalin Pithwa