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…):

CITE THIS AS:

Weisstein, Eric W. “Schwarz’s Inequality.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/SchwarzsInequality.html

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

Since

\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>

So

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

Regards,

Nalin Pithwa

VI. Countable sets: My notes.

Reference:

  1. Introduction to Topology and Modern Analysis by G. F. Simmons, Tata McGraw Hill Publications, India.
  2. Introductory Real Analysis — Kolmogorov and Fomin, Dover Publications, N.Y.(to some extent, I have blogged this topic based on this text earlier. Still, it deserves a mention/revision again).
  3. Topology : James Munkres.

The subject of this section and the next — infinite cardinal numbers — lies at the very foundation of modern mathematics. It is a vital instrument in the day to day work of many mathematicians, and we shall make extensive use of it ourselves (in our beginning studying of math ! :-)). This theory, which was created by the German mathematician Georg Cantor, also has great aethetic appeal, for it begins with ideas of extreme simplicity and develops through natural stages into an elaborate and beautiful structure of thought. In the course of our discussion, we shall answer questions which no one before Cantor’s time thought to ask, and we shall ask a question which no one can answer to this day…(perhaps !:-))

Without further ado, we can say that cardinal numbers are those used in counting, such as the positive integers (or natural numbers) 1, 2, 3, …familiar to us all. But, there is much more to the story than this.

The act of counting is undoubtedly one of the oldest of human activities. Men probably learned to count in a crude way at about the same time as they began to develop articulate speech. The earliest men who lived in communities and domesticated animals must have found it necessary to record the number of goats in the village herd by means of a pile of stones or some similar device. If the herd was counted in each night by removing one stone from the pile for each goat accounted for, then stones left over would have indicated strays, and herdsmen would have gone out to search for them. Names for numbers and symbols for them like our 1, 2, 3, …would have been superfluous. The simple yet profound idea of a one-to-one correspondence between the stones and the goats would have fully met the needs of the situation.

In a manner of speaking, we ourselves use the infinite set

N = \{ 1, 2, 3, \ldots\}

of all positive integers as “pile of stones.” We carry this set around with us as part of our intellectual equipment. Whenever we want to count a set, say, a stack of dollar bills, we start through the set N and tally off one bill against each positive integer as we come to it. The last number we reach, corresponding to the last bill, is what we call the number of bills in the stack. If this last number happens to be 10, then “10” is our symbol for the number of bills in the stack, as it also is for the number of our fingers, and for the number of our toes, and for the number of elements which can be put into one-to-one correspondence with the finite set \{ 1,2,3, \ldots, 10\}. Our procedure is slightly more sophisticated than that of the primitive savage man. We have the symbols 1, 2, 3, …for the numbers which arise in counting; we can record them for future use, and communicate them to other people, and manipulate them by the operations of arithmetic. But the underlying idea, that of the one-to-one correspondence, remains the same for us as it probably was for him.

The positive integers are adequate for our purpose of counting any non-empty finite set, and since outside of mathematics all sets appear to of this kind, they suffice for all non-mathematical counting. But in the world of mathematics, we are obliged to consider many infinite sets, such as the set of positive integers itself, the set of all integers, the set of all rational numbers, the set of all real numbers, the set of all points in a plane, and so on. It is often important to be able to count such sets, and it was Cantor’s idea to do this, and to develop a theory of infinite cardinal numbers, by means of one-to-one correspondence.

In comparing the sizes of two sets, the basic concept is that of numerical equivalence as defined in the previous section. We recall that two non-empty sets are numerically equivalent if there exists a one-to-one mapping of a set onto the other, or — and this amounts to the same thing — if there can be found a one-to-one correspondence between them. To say that two non-empty finite sets are numerically equivalent is of course to say that they have the same number of elements in the ordinary sense. If we count one of them, we simply establish a one-to-one correspondence between its elements and a set of positive integers of the form \{1,2,3, \ldots, n \} and we then say that n is the number of elements possessed by both, or the cardinal number of both. The positive integers are the finite cardinal numbers. We encounter many surprises as we follow Cantor and consider numerical equivalences for infinite sets.

The set N = \{ 1,2,3, \ldots\} of all positive integers is obviously “larger” than the set \{ 2,4,6, \ldots\} of all even positive integers, for it contains this set as a proper subset. It appears on the surface that N has “more” elements. But it is very important to avoid jumping to conclusions when dealing with infinite sets, and we must remember that our criterion in these matters is whether there exists a one-to-one correspondence between the sets (not whether one set is a proper subset or not of the other) . As a matter of fact, consider the “pairing”

1,2,3, \ldots, n, \ldots

2,4,6, \ldots, 2n, \ldots

serves to establish a one-to-one correspondence between these sets, in which each positive integer in the upper row is matched with the even positive integer (its double) directly below it, and these two sets must therefore be regarded as having the same number of elements. This is a very remarkable circumstance, for it seems to contradict our intuition and yet is based only on solid common sense. We shall see below, in Problems 6 and 7-4, that every infinite set is numerically equivalent to a proper subset of itself. Since this property is clearly not possessed by any finite set, some writers even use it as the definition of an infinite set.

In much the same way as above, we can show that N is numerically equivalent to the set of all even integers:

1, 2, 3,4, 5,6, 7, \ldots

0,2,-2,4,-4,4,6,-6, \ldots

Here, our device is start with zero and follow each even positive integer as we come to it by its negative. Similarly, N is numerically equivalent to the set of all integers:

1,2,3,4,5,6,7, \ldots

0,1,-1, 2, -2, 3, -3, \ldots

It is of considerable interest historical interest to note that Galileo had observed in the early seventeenth century that there are precisely as many perfect squares (1,4,9,16,25, etc.) among the positive integers as there are positive integers altogether. This is clear from the “pairing”:

1,2,3,4,5, \ldots

1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, \ldots

It struck him as very strange that this should be true, considering how sparsely strewn the squares are among all the positive integers. But, the time appears not to have been ripe for the exploration of this phenomenon, or perhaps he had other things on his mind; in any case, he did not follow up his idea.

These examples should make it clear that all that is really necessary in showing that an infinite set X is numerically equivalent to N is that we be able to list the elements of X, with a first, a second, a third, and so on, in such a way that it is completed exhausted by this counting off of its elements. It is for this reason that any infinite set which is numerically equivalent to N is said to be countably infinite. (Prof. Walter Rudin also, in his classic on mathematical analysis, considers a countable set to be either finite or countably infinite. ) We say that a set is countable it is non-empty and finite (in which case it can obviously be counted) or if it is countably infinite.

One of Cantor’s earliest discoveries in his study of infinite sets was that the set of all positive rational numbers (which is very large : it contains N and a great many other numbers besides) is actually countable. We cannot list the positive rational numbers in order of size, as we can the positive integers, beginning with the smallest, then the next smallest, and so on, for there is no smallest, and between any two there are infinitely many others. We must find some other way of counting them, and following Cantor, we arrange them not not in order of size, but according to the size of the sum of numerator and denominator. We begin with all positive rationals whose numerator and denominator sum add up to 2: there is only one \frac{1}{1}=1. Next, we list (with increasing numerators) all those for which this sum is 3: \frac{1}{2}, \frac{2}{1}=2. Next, all those for which the sum is 4: \frac{1}{3}, \frac{2}{2}=1, \frac{3}{1}=3. Next, all those for which this sum is 5: \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}=4. Next, all those for which this sum is 6: \frac{1}{5}, \frac{2}{4}, \frac{3}{3}=1, \frac{4}{2}=2, \frac{5}{1}=5. And, so on. If we now list all these together from the beginning, omitting those already listed when we come to them, we get a sequence like:

1, 1/2, 2, 1/3, 1/4, 2/3, 3/2, 4, 1/5, 5, \ldots

which contains each positive rational number once and only once. (There is a nice schematic representation of this : Cantor’s diagonalization process; please google it). In this figure, the first row contains all positive rationals with numerator 1, the second all with numerator 2, and so on. Our listing amounts to traversing this array of numbers in a zig-zag manner (again, please google), where of course, all those numbers already encountered are left out.

It is high time that we christened the infinite cardinal number we have been discussing, and for this purpose, we use the first letter of the Hebrew alphabet, \bf{aleph_{0}}. We say that \aleph_{0} is the number of elements in any countably infinite set. Our complete list of cardinal numbers so far is

1, 2, 3, \ldots, \aleph_{0}.

We expand this list in the next section.

Suppose now that m and n are two cardinal numbers (finite or infinite). The statement that m is less than n (written m < n) is defined to mean the following: if X and Y are sets with m and n elements, then (i) there exists a one-to-one mapping of X into Y, and (ii) there does not exist a mapping of X onto Y. Using this concept, it is easy to relate our cardinal numbers to one another by means of:

1<2<3< \ldots < \aleph_{0}.

With respect to the finite cardinal numbers, this ordering corresponds to their usual ordering as real numbers.

Regards,

Nalin Pithwa

V. Exercises: Partitions and Equivalence Relations

Reference: Topology and Modern Analysis, G. F. Simmons, Tata McGraw Hill Publications, India.

  1. Let f: X \rightarrow Y be an arbitrary mapping. Define a relation in X as follows: x_{1} \sim x_{2} means that f(x_{1})=f(x_{2}). Prove that this is an equivalence relation and describe the equivalent sets.

Proof : HW. It is easy. Try it. 🙂

2. In the set \Re of real numbers, let x \sim y means that x-y is an integer. Prove that this is an equivalence relation and describe the equivalence sets.

Proof: HW. It is easy. Try it. 🙂

3. Let I be the set of all integers, and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m — symbolized by a \equiv b \pmod {m} — if a-b is exactly divisible by m, that is, if a-b is an integral multiple of m. Show that this is an equivalence relation, describe the equivalence sets, and state the number of distinct equivalence sets.

Proof: HW. It is easy. Try it. 🙂

4. Decide which one of the three properties of reflexivity, symmetry and transitivity are true for each of the following relations in the set of all positive integers: m \leq n, m < n, m|n. Are any of these equivalence relations?

Proof. HW. It is easy. Try it. 🙂

5. Give an example of a relation which is (a) reflexive, but not symmetric or transitive. (b) symmetric but not reflexive or transitive. (c) transitive but not reflexive or symmetric (d) reflexive and symmetric but not transitive (e) reflexive and transitive but not symmetric. (f) symmetric and transitive but not reflexive.

Solutions. (i) You can try to Google (ii) Consider complex numbers (iii) there are many examples given in the classic text “Discrete Mathematics” by Rosen.

6) Let X be a non-empty set and \sim a relation in X. The following purports to be a proof of the statement that if this relation is symmetric and transitive, then it is necessarily reflexive: x \sim y \Longrightarrow y \sim x ; x \sim y and y \sim x \Longrightarrow x; therefore, x \sim x for every x. In view of problem 5f above, this cannot be a valid proof. What is the flaw in the reasoning? 🙂

7) Let X be a non-empty set. A relation \sim in X is called circular if x \sim y and y \sim x \Longrightarrow z \sim x, and triangular if x \sim y and x \sim z \Longrightarrow y \sim z. Prove that a relation in X is an equivalence relation if and only if it is reflexive and circular if and only if it is reflexive and triangular.

HW: Try it please. Let me know if you need any help.

Regards,

Nalin Pithwa.

PS: There are more examples on this topic of relations in Abstract Algebra of I. N. Herstein and Discrete Mathematics by Rosen.

IV. Product of Sets: Exercises

Reference: Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Publications, India.

Problems:

I) The graph of a mapping f: X \rightarrow Y is a subset of the product X \times Y. What properties characterize the graphs of mappings among all subsets of X \times Y?

Solution I: composition.

II) Let X and Y be non-empty sets. If A_{1} and A_{2} are subsets of X, and Y_{1} and Y_{2} are subsets of Y, then prove the following

(a) (A_{1} \times B_{1}) \bigcap (A_{2} \times B_{2}) = (A_{1} \bigcap A_{2}) \times (B_{1} \bigcap B_{2})

(b) (A_{1} \times B_{1}) - (A_{2} \times B_{2}) = (A_{1}-A_{2}) \times (B_{1}-B_{2}) \bigcup (A_{1} \bigcap A_{2}) \times (B_{1}-B_{2}) \bigcup (A_{1}-A_{2}) \times (B_{1} \bigcap B_{2})

Solution IIa:

TPT: (A_{1} \times B_{1}) \times (A_{2} \times B_{2}) = (A_{1} \bigcap A_{2}) \times (B_{1} \bigcap B_{2})

This is by definition of product and the fact that the co-ordinates are ordered and the fact that A_{1} \subseteq X, A_{2} \subseteq X, B_{1} \subseteq Y, and B_{2} \subseteq Y.

Solution IIb:

Let (a_{i}, b_{j}) \in A_{1} \times B_{1}, but (a_{i}, b_{j}) \notin A_{2} \times B_{2}. So, the element may belong to (A_{1}-A_{2}) \times (B_{1} - B_{2}) or it could happen that it belongs to A_{1} \times A_{1}, but to (B_{1}-B_{2}) (here we need to remember that the element is ordered); so, also it could be the other way: it could belong to (A_{1}-A_{2}) but to (B_{1} \bigcap B_{2}) also. The same arguments applied in reverse establish the other subset inequality. Hence, done.

III) Let X and Y be non-empty sets, and let A and B be rings of subsets of X and Y, respectively. Show that the class of all finite unions of sets of the form A \times B with A \in \bf{A} and B \in \bf{B} is a ring of subsets of X \times Y.

Solution III:

\bigcup_{i=1}^{n}A_{i} \times B_{i} = \bigcup_{i=1}^{n}A_{i} \times \bigcup_{i=1}^{n}B_{i}.

From question IIb above, the difference of any two pairs of sets is also in X \times Y.

Hence, done.

Regards,

Nalin Pithwa

IV. Product of Sets

Reference: Topology and Modern Analysis — G. F. Simmons, Tata McGraw Hill, India.

We shall often have occasion to weld together the sets of a given class into a single new set called their product (or their Cartesian product). The ancestor of this concept is the coordinate plane of analytic geometry, that is, a plane equipped with the normal rectangular coordinate axes. We give a brief description of this fundamental idea with a view to paving the way for our discussion of product of sets in general.

First, a few preliminary comments about the real line. We have already used this term several times before without any explanations, and of course what we mean by it is an ordinary geometric line whose points have been identified with — or coordinatized by — the set R of all real numbers. We use the letter R to denote the real line as well as the set of all real numbers, and we often speak of real numbers as if they were points on the real line, and of points on the real line as if they were real numbers. Let no be deceived into thinking that the real line is a simple thing, for its structure is exceedingly intricate. Our present view of it, however, is as naive and uncomplicated as the picture of it. Generally speaking, we assume that the reader is familiar with the simpler properties of the real line — those relating to inequalities (see problems section) and the basic algebraic operations of addition, subtraction, multiplication and division. One of the most significant facts about the real number system is perhaps less well known. This is the so-called least upper bound property. It asserts that every non empty set of real numbers which has an upper bound has a least upper bound. It is an easy consequence of this fact that a non empty set of real numbers which has a lower bound has a greatest lower bound. All these matters are developed rigorously on the basis of a small number of axioms, and detailed treatments can often be found in books on elementary abstract algebra.

To construct the coordinate plane, we now proceed as follows. We take two identical replicas of the real line, which we call the x axis and the y axis, and paste them on a plane at right angles to one another in such a way that they cross at the zero point on each. We know that usual picture. Now, let P be a point in the plane. We project P perpendicularly onto points Px and Py on the axes. If x and y are the coordinates of Px and Py on their respective axes, this process leads us from the point P to the uniquely determined ordered pair (x,y) of real numbers, where x and y are called the x coordinate and y coordinate of the point P. We can reverse the process, and starting with the ordered pair of real numbers, we can recapture the point. This is the manner in which we establish the familiar one-to-one correspondence between points P in the plane and ordered pairs (x,y) of real numbers. In fact, we think of a point in the plane (which is a geometric object) and its corresponding ordered pair of real numbers (which is an algebraic object) as being — to all intents and purposes — identical with one another. The essence of analytic geometry lies in the possibility of this exploiting this identification by using algebraic tools in geometric arguments and giving geometric interpretations to algebraic calculations.

The conventional attidute towards the coordinate plane in analytic geometry is that the geometry is the focus of interest and the algebra of ordered pairs is only a convenient tool. Here, we reverse this point of view. For us, the coordinate plane is defined to be the set of all ordered pairs (x,y) of real number. We can satisfy our desire for visual images by using the usual picture of the plane and by calling such an ordered pair a point, but this geometric lnaguage is more a convenience than a necessity.

Our notation for the coordinate plane is R \times R or R^{2}. This symbolism reflects the idea that the coordinate plane is the result of multiplying together two replicas of the real line R.

It is perhaps necessary to comment on one possible source of misunderstanding. When we speak of R^{2} as a plane, we do so only to establish an intuitive bond with the reader’s previous experience in analytic geometry. Our present attitude is that R^{2} is a pure set and has no structure whatsoever, because no structure has as yet been assigned to it. We remarked earlier (with deliberate vagueness) that a space is a set to which has been added some kind of algebraic or geometric structure. Later, we shall convert the set R^{2} into the space of analytic geometry by defining the distance between any two points (x_{1}, y_{1}) and (w_{2}, y_{2}) to be

\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}

This notion of distance endows the set R^{2} with a certain “spatial” character, which we shall recognize by calling the resulting space the Euclidean plane instead of the coordinate plane.

We assume that the reader is fully acquainted with the way in which the set C of all complex numbers can be identified (as a set) with the coordinate plane R^{2}. If z is a complex number, and if z has the standard form x+iy where x and y are real numbers, then we identify z with the ordered pair (x,y) and thus, with an element of R^{2}. The complex numbers, however, are much more than merely a set. They constitute a number system with the operations of addition, multiplication, conjugation, etc. When the coordinate plane R^{2} is thought of as consisting of complex numbers and is enriched by the algebraic structure it acquires in this way, it is called the complex plane. The letter C is used to denote either the set C of all complex numbers or the complex plane. Later, we shall make a space out of the complex plane.

Suppose now that X_{1} and X_{2} are two non empty sets. By analogy with our above discussion, their product X_{1} \times X_{2} is defined by the set of all ordered pairs (x_{1}, x_{2}) where x_{1} is in X_{1} and x_{2} is in X_{2}. In spite of the arbitrary nature of X_{1} and X_{2}, their product can be represented by a picture similar to the XY plane. The term product is applied to this set, and it is thought of as a result of multiplying together the sets X_{1} and X_{2} for the following reason: if X_{1} and X_{2} are finite sets with m and n elements, then clearly X_{1} \times X_{2} has mn elements. If f: X_{1} \rightarrow X_{2} is a mapping with domain X_{1} and range X_{2}, its graph is a subset of X_{1} \times X_{2} which consists of all ordered pairs of the form (x_{1}, f(x_{1})). We observe that this is an appropriate generalization of the concept of a graph of a function as it occurs in elementary mathematics.

This definition of the product of two sets extends easily to the definiion of product of n sets for any positive integer n. If X_{1}, X_{2}, X_{3}, \ldots, X_{n} are n sets where n is any positive integer, then their product X_{1} \times X_{2} \times X_{3} \times \ldots \times X_{n} is the set of all ordered tuples (x_{1}, x_{2}, \ldots, x_{n}) where x_{i} is in the set X_{i} for each subscript i. If the X_{i}‘s are all replicas of a single set X, that is, if

X_{1} = X_{2} = \ldots = X_{n} = X,

then their product is usually denoted by the symbol X^{n}.

These ideas specialize directly to yield the important sets R^{n} and C^{n}. R is just R, the real line; and R^{2} is the coordinate plane; R^{3} is the set of all ordered triples of real numbers — the set which underlies solid analytic geometry, and we assume that the reader is familiar with the manner in which this set arises, through the introduction of a rectangular coordinate system into three dimensional space. We can draw pictures here, just as in the case of the coordinate plane and use geometric language here as much as we please, but it must be understood that the mathematics of this set is the mathematics of ordered set of triples of real numbers and that pictures are merely an aid to the intuition. Once we fully this grasp this point of view, there is no difficulty whatsoever in advancing at once to the study of the set R^{n}, study of n-tuples of real numbers for any positive integer n. It is quite true that when n is greater than 3, it is no longer possible to draw the same kind of intuitively rich pictures, but at worst this is merely an inconvenience. We can and do continue to use suggestive geometric language, so all is not lost. The set C^{n} is defined similarly; it is the set of all ordered n-tuples (z_{1}, z_{2}, \ldots, z_{n}) of complex numbers. Both R^{n} and C^{n} are of fundamental importance in analysis and topology.

We emphasized that for the present the coordinate plane is to be considered merely as a set, but not as a space. Similar remarks apply to R^{n} and C^{n}. In due conrse, we shall impart form and content to each of these sets by suitable definitions. We shall convert them into Euclidean and n-unitary space which underlie and motivate so many developments in pure mathematics, and we shall explore some aspects of their algebraic and topological structures, But, as of now — and this is the point of view we insist on — they do not have any structure at all; they are merely sets.

As the reader doubtless suspects, we need not consider only products of finite classes of sets. The needs of topology compel us to extend these ideas to arbitrary classes of sets.

We defined the product X_{1} \times X_{2} \times \ldots \times X_{n} to be the set of all ordered n-tuples (x_{1}, x_{2}, \ldots, x_{n}) such that x_{i} is in X_{i} for each subscript i. To see how to extend this definition, we reformulate it as follows: We have an nidex set I, consisting of the integers from 1 to n, and corresponding to each index (or subscript) i we have a non-empty set X_{i}. The n-tuple (x_{1}, x_{2}, x_{3}, \ldots, x_{n}) is simply a function (call it x) defined on the index set I, with the restriction that its value x(i)=x_{i} is an element of the set X_{i} for each i in I. Our point of view here is that the function x is completely determined by and is essentially equivalent to the array (x_{1}, x_{2}, x_{3}, \ldots, x_{n}) of its values.

The way is now open for the definition of products in their full generality. Let \{ X_{i}\} be a non-empty class of non-empty sets, indexed by the elements i of an index set I. The sets X_{i} need not be different from one another; indeed, it may happen that they are all identical replicas of a single set, distinguished only different indices. The product of the sets X_{i}, written P_{i \in I}X_{i} is defined to be the set of all functions x defined on I such that x(i) is an element of the set X_{i} for each index i. We call X_{i} the ith coordinate set. When there can be no misunderstanding about the index set, the symbol P_{i \in I}X_{i} is often abbreviated to P_{i}X_{i}. The definition we have just given requires that each coordinate set be non-empty before the product can be formed. It will be useful if we extend this definition slightly by agreeing that if any of the X_{i}‘s are empty, then P_{i}X_{i} is also empty.

This approach to the idea of a product of a class of sets, by means of functions defined on the index set is useful mainly in giving the definition. In practice, it is much more convenient to use the subscript notation x_{i} instead of the function notation x(i). We then interpret the product P_{i}X_{i} as made up of elements x, each of which is specified by the exhibited array \{ x_{i}\} of it values in the respective coordinate sets X_{i}. We call x_{i} the ith coordinate of the element x = \{ x_{i}\}.

The mapping p_{i} of the product P_{i}X_{i} onto its ith coordinate set X_{i} which is defined by p_{i}(x) = x_{i} — that is, the mapping whose value at an arbitrary element of the product is the ith coordinate of that element — is called the projection onto the ith coordinate set. The projection p_{i} selects the ith coordinate of each element in its domain. There is clearly one projection for each element of the index set I, and the set of all projections plays an important role in the general theory of topological spaces.

We will continue with exercises on this topic in a later blog.

Regards,

Nalin Pithwa

Notes II: Sets and Functions:

Reference I : Topology and Modern Analysis, G F Simmons, Tata McGraw Hill.

III. Functions:

Many kinds of functions occur in topology in a wide variety of situations. In our work, we shall need the full power of the general concept of a function, and since, its modern meaning is much broader and deeper than its elementary meaning, we discuss this concept in considerable detail and develop its main abstract properties.

Let us begin with a brief inspection of some simple examples. Consider the elementary function

y=x^{2}

of the real variable x. What do we have in mind when we call this a function and say that y is a function of x? In a nutshell, we are drawing attention to the fact that each real number x has a specific real number y linked to it, which can be calculated according to the rule (or law of correspondence) given by the formula. We have here a process which applied to a real number x does something to it (squares it) to produce another number y (the square of x). Similarly,

y =  x^{2}-3x and y = (x^{2}+1)^{-1}

are two other simple functions of the real variable x, and each is given by a rule in the form of an algebraic expression which specifies the exact manner in which the value of y depends on the value of x.

The rules for the functions we have just mentioned are expressed by formulas. In general, this is possible for functions of a very simple kind or for those which are sufficiently important to deserve a special symbol of their own. Consider for instance the function of the real variable x defined as follows: for each real number x, write x as an infinite decimal (using the scheme of decimal expansion in which infinite chains of 9s are avoided — in which for example, 1/4 is represented as 0.250000….rather than by 0.24999….); then, let y be the 59th digit after the decimal point. There is of course no standard formula for this but nevertheless it is a perfectly respectable function whose rule is given by a verbal description. On the other hand, the function y=\sin{x} of the real variable x is so important that its rule, though fully as complicated as the one just defined is assigned the special symbol sin. When discussing functions in general, we work to allow for all sorts of rules and to talk about them all at once, so we simply employ non-committal notations like y=f(x), y=g(z), and so on.

Each of the functions mentioned above is defined for all real numbers x. The example y = \frac{1}{x} shows that this restriction is much too severe, for this function is defined only for non zero values of x. Similarly, y = \log{x} is defined only for positive values of x and y = \arcsin{x} only for values of x in the interval [-1,1]. Whatever our conception of a function may be, it should certainly be broad enough to include examples like these, which are defined only for some values of the real variable x.

In real analysis, the notion of function is introduced in the following way. Let X be any non-empty set of real numbers. We say that a function y=f(x) is defined on X if the rule f associates a definite real number y with each real number x in X. The specific nature of the rule f is totally irrelevant to the concept of a function. The set X is called the domain of the given function, and the set Y of all the values it assumes is called its range. If we speak of complex numbers here instead of real numbers, we have the notion of function as it is used in complex analysis.

This point of view towards functions is actually more general than is needed for aims of analysis, but it isn’t nearly general enough for our purposes. The sets X and Y above were taken to be sets of numbers. If we now remove even this restriction and allow X and Y to be completely arbitrary non-empty sets, then we arrive at the most inclusive concept of a function. By way of illustration, suppose that X is the set of all squares in a plane and that Y is the set of all circles in the same plane. We can define a function y=f(x) by requiring that the rule f associate with each square x that circle y which is inscribed in it. In general, there is no need at all for either X or Y to be a set of numbers. All that is really necessary for a function is two non-empty sets X and Y and a rule f which is meaningful and unambiguous in assigning to each element x in X a specific element y in Y.

With these preliminary descriptive remarks, we now turn to the rather abstract but very precise ideas they are intended to motivate.

A function consists of three objects: two non-empty sets X and Y (which may be equal but need not be) and a rule f which assigns to each element x in X a single fully determined element y in Y. The y which corresponds in this way to a given x is usually written f(x), and is called the image of x under the rule f, or the value of f at the element x. (It is fun to draw some figures here). This notation is supposed to be suggestive of the idea that the rule f takes the element x and does something to it to produce the element y = f(x). The rule f is often called a mapping or transformation or operator to amplify this concept of it. We then think of f as mapping x’s to y’s, or transforming x’s to y’s, or operating on x’s to produce y’s. The set X is called the domain of the function, and the set of all f(x)’s for all x’s in X is called its range. A function whose range consists of just one element is called a constant function.

We often denote by f: X \rightarrow Y the function with rule f, domain X and range contained in Y. This notation is useful because the essential parts of the function are displayed in a manner which emphasizes that it is a composite object, the central thing being the rule or mapping f. You can try drawing a figure depicting a convenient way of picturing this function. (these notes don’t have my diagrams from the reference book) On the left, X and Y are different sets, and on the right they are equal — in which case we usually refer to f as a mapping of X into itself. If it is clear that from the context what the sets X and Y are, or if there is no real need to specify them explicitly, it is common practice to identify the function f: X \rightarrow Y with the rule f, and to speak of f alone as if it were the function under consideration (without mentioning the sets X and Y).

It sometimes happens that two perfectly definite sets X and Y are under discussion and that a mapping of X into Y arises which has no natural symbol attached to it. If there is no necessity to invent a symbol for this mapping and if it is quite clear what the mapping is, it is often convenient to designate it by x \rightarrow y. Accordingly, the function y=x^{2} mentioned in the beginning of this section can be written as x \rightarrow x^{2} or x \rightarrow y where y is understood to be the square of x.

A function f is called an extension of a function g (and g is called a restriction of f) if the domain of f contains the domain of g and f(x) = g(x) for each x in the domain of y.

Most of mathematical analysis, both classical and modern, deals with functions whose values are real numbers or complex numbers. This is also true of those parts of topology which are concerned with the foundations of analysis. If the range of a function consists of real numbers, we call it a real function, similarly, a complex function is one whose range consists of complex numbers. Obviously, every real function is also complex. We lay very heavy emphasis on real and coomplex functions through out our work.

As a matter of usage, we generally prefer to reserve the term function for real or complex functions and to speak of mappings when dealing with functions whose values are not necessarily numbers.

Consider a mapping f: X \rightarrow Y. When we call f a mapping of X into Y, we mean to suggest by this that the elements f(x) — as x varies over all the elements of X — need not fill up Y; but if it definitely does happen that the range of f equals Y, or if we specifically want to assume this, then we call f a mapping of X onto Y. If two different elements in X always have different images under f, then we call f a one=to-one mapping of X into Y. If f: X \rightarrow Y is both onto and one-to-one, then we can define its inverse mapping f^{-1}: X \rightarrow Y as follows: for each y in Y, we find that unique element x in X such that f(x)=y 9 x exists and is unique since f is onto and one-to-one); we then define x to be f^{-1}(y). The equation x = f^{-1}(y) is the result of solving y=f(x) for x in just the same way as x = \log{y} is the result of solving e^{x} for x. Figure 7 illustrates the concept of the inverse of a mapping.

If f is a one-to-one mapping of X onto Y, it will sometimes be convenient to subordinate the conception of f as a mapping sending x’s over to y’s and to emphasize its role as a link between x’s an y’s. Each x has linked to it (or has corresponding to it) exactly one x = f^{-1}(y). When we focus our attention on this aspect of a mapping which is one-to-one onto correspondence between X and Y, and f^{-1} is a one-to-one correspondence between Y and X.

Now, consider an arbitrary mapping f: X \rightarrow Y. The mapping f which sends each element of X over to an element of Y induces the following important set mappings. If A is a subset of X, then the image f(A) is the subset of Y defined by

f^{-1}(B) = \{ x : f(x) \in \hspace{0.1in} B\}

and the second set mappings pull each B back to its corresponding f^{-1}(B). It is often essential for us to know how these set mappings behave with respect to set inclusion and operations on sets. We develop most of their significant features in the following two paragraphs.

The main properties of the first set mapping are:

f(\phi) = \phi

f(X) \subseteq Y

A_{1} \subseteq A_{2} \Longrightarrow f(A_{1}) \subseteq f(A_{2})

f(\bigcup_{i})A_{i} = \bigcup_{i}f(A_{i})

f(\bigcap_{i}A_{i}) \subseteq \bigcup_{i}f(A_{i})….call these relations I.

The reader should convince himself of the truth of these statements. For instance, to prove (i) we would have to prove first that f(\bigcup_{i}A_{i}) is a subset of \bigcup_{i}f(A_{i}), and second that \bigcup_{i}f(A_{i}) is a subset of f(\bigcup_{i}A_{i}). A proof of the first of these set inclusions might run as follows: an element in f(\bigcup_{i}A_{i}) is the image of some element in \bigcup_{i}A_{i}, therefore, it is the image of an element in some A_{i}, therefore it is some f(A_{i}) and so finally it is in \bigcup_{i}f(A_{i}). The irregularities and gaps which the reader will notice in the above statements are essential features of this set mapping. For example, the image of an intersection need not equal the intersection of the images, because two disjoint sets can easily have images which are not disjoint. Furthermore, without special assumpitions (see Problem 6), nothing can be said about the relation between f(A)^{'} and f(A^{'}).

The second set mapping is much better behaved. Its properties are satisfyingly complete, and can be stated as follows:

f^{-1}(\phi) = \phi and f^{-1}(Y) =X;

B_{1} \subseteq B_{2} \Longrightarrow f^{-1}(B_{1}) \subseteq f^{-1}(B_{2})

f^{-1}(\bigcup_{i}B_{i}) \Longrightarrow \bigcup_{i}f^{-1}(B_{i})….(2)

f^{-1}(\bigcap_{i}B_{i}) = \bigcap_{i}f^{-1}(B_{i})….(3)

f^{-1}(B^{'}) = f^{-1}(B)^{'}….(4)

Again, the reader should verify each of these statements for himself.

We discuss one more concept in this section, that of the multiplication or composition of mappings. If y= f(x)=x^{2}+1 and z = g(y) = \sin{y}

then these two functions can be put together to form a single function defined by x = (gf)(x) = g(x^{2}+1)=\sin{(x^{2}+1)}. One of the most important tools of calculus (the chain rule) explains how to differentiate functions of this kind. This manner of multiplying functions together is of basic importance for us as well, and we formulate it in general as follows. We define the product of these mappings, denoted by gf: X \rightarrow Z by (gf)(x) = g(f(x)). In words, an element x in X is taken by f to the element f(x) in Y,and then g maps f(x) to g(f(x)) in Z. Figure 8 is a picture of this process. We observe that the two mappings involved here are not entirely arbitrary, for the set Y which contains the range of the first equals the domain of the second. More generally, the product of two mappings is meaningful whenever the range of the first is contained in the domain of the second. We have regarded f as the first mapping and y as the second, and in forming their product gf, their symbols have gotten turned around. This is a rather unpleasant phenomenon, for which we blame the occasional perversity of mathematical symbols. Perhaps it will help the reader to keep this straight in his mind if he will remember to read the product gf from right to left: first apply f, then g.

Problems:

To be continued next blog.

Regards,

Nalin Pithwa

Problem Set based on VII. Complete Metric Spaces

Problem 1.

Prove that the limit f(t) of a uniformly convergent sequence of functions \{ f_{n}(t)\} continuous on [a,b] is itself a function continuous on [a,b].

Hint: Clearly,

|f(t)-f(t_{0})| \leq |f(t) - f_{n}(t)|+|f_{n}(t) - f_{n}(t_{0})|+|f_{n}(t_{0}) - f(t_{0})| where t, t_{0} \in [a,b].

Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large n. Then, use the continuity of f_{n}(t) to make the second term small for t sufficiently close to t_{0}.

Problem 2.

Prove that if R is complete, then the intersection \bigcap_{n=1}^{\infty}S_{n} figuring in the theorem 2 consists of a single point. Theorem 2 is recalled here: Nested sphere theorem: A metric space R is complete if and only if nested sequence S_{n} = \{ S[x_{n}, r_{n}]\} of closed spheres in R such that t_{n} \rightarrow 0 as n \rightarrow \infty has a non empty intersection \bigcap_{n=1}^{\infty}S_{n}.

Problem 3.

Prove that the space m is complete. Recall: Consider the set of all bounded infinite sequences of real numbers x = (x_{1}, x_{2}, \ldots, x_{k}, \ldots) and let \rho(x,y) = \sup_{k}|x_{k}-y_{k}|. This is a metric space which we denote by m.

Problem 4:

By the diameter of a subset A of a metric space R is meant the number d(A) = \sup_{x, y \in A} \rho(x,y)

Suppose R is complete and let \{ A_{n}\} be a sequence of closed sets of R nested in the sense that

A_{1} \supset A_{2} \supset \ldots \supset A_{n} \supset \ldots

Suppose further that

\lim_{n \rightarrow \infty} d(A_{n})=0.

Prove that the intersection \bigcap_{n=1}^{\infty}A_{n} is non empty.

Problem 5.

A subset A of a metric space R is said to be bounded if its diameter d(A) is finite. Prove that the union of a finite number of bounded sets is bounded.

Problem 6.

Give an example of a complete metric space R and a nested sequence \{ A_{n}\} of closed subsets of R such that

\bigcap_{n=1}^{\infty}A_{n} = \phi.

Reconcile this example with Problem 4 here.

Problem 7.

Prove that a subspace of a complete metric space R is complete if and only if it is closed.

Problem 8.

Prove that the real line equipped with the distance metric function

\rho(x,y) = |\arctan{x} - \arctan{y}| is an incomplete metric space.

Problem 9.

Give an example of a complete metric space homeomorphic to an incomplete metric space.

Hint: Consider the following example we encountered earlier: The function y = f(x) = \frac{2}{\pi}\arctan{x} establishes a homeomorphism between the whole real line (-\infty, \infty) and the open interval (-1,1).

Comment: Thus, homeomorphic metric spaces can have different metric properties.

Problem 10:

Carry out the program discussed in the last sentence of the following example:

Ir R is the space of all rational numbers, then R^{*} is the space of all real numbers, both equipped with the distance \rho(x,y) = |x-y|. In this way, we can “construct the real number system.” However, there still remains the problem of suitably defining sums and products of real numbers and verifying that the usual axioms of arithmetic are satisfied.

Hint: If \{ x_{n}\} and \{ y_{n}\} are Cauchy sequences of rational numbers serving as “representatives” of real numbers x^{*} and y^{*} respectively, define x^{*} = y^{*} as the real number with representative \{ x_{n} + y_{n}\}.

I will post the solutions in about a week’s time.

Regards,

Nalin Pithwa.

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Cauchy’s Mean Value Theorem and the Stronger Form of l’Hopital’s Rule

Reference: G B Thomas, Calculus and Analytic Geometry, 9th Indian Edition. 

The stronger form of l’Hopital’s rule is as follows:

Suppose that f(x_{0})=g(x_{0})=0 and the functions f and g are both differentiable on an open interval (a,b) that contains the point x_{0}. Suppose also that g^{'} \neq 0 at every point in (a,b) except possibly x_{0}. Then,

\lim_{x \rightarrow x_{0}} \frac{f(x)}{g(x)} = \lim_{x \rightarrow x_{0}}\frac{f^{'}(x)}{g^{'}(x)}…call this I, provided the limit on the right exists.

Remarks:

The proof of the stronger from of l’Hopital’s rule in based on Cauchy’s mean value theorem, a mean value theorem that involves two functions instead of one. We prove Cauchy’s theorem first and then show how it leads to l’Hopital’s rule.

Cauchy’s Mean Value Theorem:

Suppose the functions f and g are continuous on [a,b] and differentiable through out (a,b) and suppose also that g^{'} \neq 0 through out (a,b). Then, there exists a number c in (a,b) at which

\frac{f^{'}(c)}{g^{'}(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}.

(Note this becomes the ordinary mean value theorem when g(x)=x).

Proof of Cauchy’s Mean Value theorem:

We apply the ordinary mean value theorem twice. First, we use it to show that g(b) \neq g(a). Because if g(b)=g(a), then the ordinary Mean Value theorem says that

g^{'}(c) = \frac{g(b)-g(a0}{b-a}=0 for some c between a and b. This cannot happen because g^{'}(x) \neq 0 in (a,b).

We next apply the Mean Value Theorem to the function

F(x) = f(x)-f(a) - \frac{f(b)-f(a)}{g(b)-g(a)}(g(x)-g(a))

This function is continuous and differentiable where f and g are, and note that F(b)=F(a)=0. Therefore, by the ordinary mean value theorem, there is a number c between a and b for which F^{'}(c)=0. In terms of f and g, this says

F^{'}(c) = f^{'}(c) - \frac{f(b)-f(a)}{g(b)-g(a)}(g^{'}(c)) = 0

or \frac{f^{'}(c)}{g^{'}(c)} = \frac{f(b)-f(a)}{g(b)-g(a)} which is equation II above.

Proof of the stronger form of L’Hopital’s Rule:
We first establish equation I for the case \lim x \rightarrow x_{0}^{+}. The method needs almost no change to apply to the case \lim x \rightarrow x_{0}^{-}, and the combination of these two cases establishes the result.

Suppose that x lies to the right of x_{0}. Then, g^{'}(x) \neq 0 and we can apply Cauchy’s Mean Value theorem to the closed interval from x_{0} to x. This produces a number c between x and x_{0} such that

\frac{f^{'}(c)}{g^{'}(c)} = \frac{f(x)-f(x_{0})}{g(x)-g(x_{0})}

But, f(x_{0})=g(x_{0})=0 so

That \frac{f^{'}(c)}{g^{'}(c)}= \frac{f(x)}{g(x)}

As x approaches x_{0}, c approaches x_{0} as it lies between x and x_{0}. Therefore,

\lim_{x \rightarrow x_{0}^{+}} \frac{f(x)}{g(x)} = \lim_{x \rightarrow x_{0}^{+}}  \frac{f^{'}(c)}{g^{'}(c)} = \lim_{x \rightarrow x_{0}^{+}} \frac{f^{'}(x)}{g^{'}(x)}.

This establishes l’Hopital’s Rule for the case where approaches x_{0} from right. The case where x approaches x_{0} from the left is proved by applying Cauchy’s Mean Value Theorem to the closed interval [x,x_{0}] when x < x_{0}.

QED.

Regards,

Nalin Pithwa