# A First Course in Functional Analysis: Theory and Applications

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## Overview

A comprehensive introduction to functional analysis, starting from the fundamentals and extending into theory and applications across multiple disciplines.

## Product Details

ISBN-13: 9781783083244 Anthem Press 11/01/2014 Reprint 486 6.00(w) x 8.90(h) x 1.10(d)

Rabindranath Sen is a retired professor and former head of the Department of Applied Mathematics at the University of Calcutta.

#### Theory and Applications

By Rabindranath Sen

#### Wimbledon Publishing Company

ISBN: 978-0-85728-320-7

CHAPTER 1

PRELIMINARIES

In this chapter we recapitulate the mathematical preliminaries that will be relevant to the development of functional analysis in later chapters. This chapter comprises six sections. We presume that the reader has been exposed to an elementary course in real analysis and linear algebra.

1.1 Set

The theory of sets is one of the principal tools of mathematics. One type of study of set theory addresses the realm of logic, philosophy and foundations of mathematics. The other study goes into the highlands of mathematics, where set theory is used as a medium of expression for various concepts in mathematics. We assume that the sets are 'not too big' to avoid any unnecessary contradiction. In this connection one can recall the famous 'Russell's Paradox' (Russell, 1959). A set is a collection of distinct and distinguishable objects. The objects that belong to a set are called elements, members or points of the set. If an object a belongs to a set A, then we write a [member of] A. On the other hand, if a does not belong to A, we write a [??] A. A set may be described by listing the elements and enclosing them in braces. For example, the set A formed out of the letters a, a, a, b, b, c can be expressed as A = {a, b, c}. A set can also be described by some defining properties. For example, the set of natural numbers can be written as N = {x : x, a natural number} or {x|x, a natural number}. Next we discuss set inclusion. If every element of a set A is an element of the set B, A is said to be a subset of the set B or B is said to be a superset of A, and this is denoted by A [??] B or B [??] A. Two sets A and B are said to be equal if every element of A is an element of B and every element of B is an element of A – in other words if A [??] B and B [??] A. If A is equal to B, then we write A = B. A set is generally completely determined by its elements, but there may be a set that has no element in it. Such a set is called an empty (or void or null) set and the empty set is denoted by Φ (Phi). Φ [subset] A; in other words, the null set is included in any set A – this fact is vacuously satisfied. Furthermore, if A is a subset of B, [A ≠ Φ and A ≠ B then A is said to be a proper subset of B (or B is said to properly contain A). The fact that A is a proper subset of B is expressed as A [subset] B. Let A be a set. Then the set of all subsets of A is called the power set of A and is denoted by P(A). If A has three elements like letters p, q and r, then the set of all subsets of A has 8(= 23) elements. It may be noted that the null set is also a subset of A. A set is called a finite set if it is empty or it has n elements for some positive integer n; otherwise it is said to be infinite. It is clear that the empty set and the set A are members of P (A). A set A is called denumerable or enumerable if it is in one-to-one correspondence with the set of natural numbers. A set is called countable if it is either finite or denumerable. A set that is not countable is called uncountable.

We now state without proof a few results which might be used in subsequent chapters:

(i) An infinite set is equivalent to a subset of itself.

(ii) A subset of a countable set is a countable set.

The following are examples of countable sets: a) the set J of all integers, b) the set Q of all rational numbers, c) the set P of all polynomials with rational coefficients, d) the set all straight lines in a plane each of which passes through (at least) two different points with rational coordinates and e) the set of all rational points in Rn.

Examples of uncountable sets are as follows: (i) an open interval [a, b], a closed interval [a, b] where a ≠ b, (ii) the set of all irrational numbers. (iii) the set of all real numbers. (iv) the family of all subsets of a denumerable set.

1.1.1 Cardinal numbers

Let all the sets be divided into two families such that two sets fall into one family if and only if they are equivalent. This is possible because the relation ~ between the sets is an equivalence relation. To every such family of sets, we assign some arbitrary symbol and call it the cardinal number of each set of the given family. If the cardinal number of a set A is α, A = α or card A = α The cardinal number of the empty set is defined to be 0 (zero). We designate the number of elements of a nonempty finite set as the cardinal number of the finite set. We assign N0 to the class of all denumerable sets and as such N0 is the cardinal number of a denumerable set. c, the first letter of the word 'continuum' stands for the cardinal number of the set [0, 1].

1.1.2 The algebra of sets

In the following section we discuss some operations that can be performed on sets. By universal set we mean a set that contains all the sets under reference. The universal set is denoted by U. For example, while discussing the set of real numbers we take R as the universal set. Once again for sets of complex numbers the universal set is the set C of complex numbers. Given two sets A and B, the union of A and B is denoted by A [union] B and stands for a set whose every element is an element of either A or B (including elements of both A and B). A [union] B is also called the sum of A and B and is written as A + B. The intersection of two sets A and B is denoted by A [intersection] B, and is a set, the elements of which are the elements common to both A and B. The intersection of two sets A and B is also called the product of A and B and is denoted by A x B. The difference of two sets A and B is denoted by A-B and is defined by the set of elements in A which are not elements of B. Two sets A and B are said to be disjoint if A [intersection] B = Φ. If A [??] B, B - A will be called the complement of A with reference to B. If B is the universal set, Ac will denote the complement of A and will be the set of all elements which are not in A.

Let A, B and C be three non-empty sets. Then the following laws hold true:

1. Commutative laws

A [union] B = B [union] A and A [intersection] B = B [intersection] A

2. Associative laws

We have a finite number of sets

A [union] (B [union] C) = (A [union] B) [union] C and (A [intersection] B) [intersection] C = A [intersection] (B [intersection] C)

3. Distributive laws

A [intersection] (B [union] C) = (A [intersection] B) [union] (A [intersection] C) (A [union] B) [intersection] C = (A [intersection] C) [union] (A [intersection] B)

4. De Morgan's laws

(A [union] B)c = (Ac [intersection] Bc) and (A [intersection] Bc) = (A [intersection] B)

Suppose we have a finite class of sets of the form {A1, A2A3 ,..., An} then we can form A1 [union] A2 [union] A3 [union] ... An and A1 [intersection] A2 [intersection] A3 [intersection] ... An. We can shorten the above expression by using the index set I = {1, 2, 3, ..., n}. The above expressions for union and intersection can be expressed in short by [union]i [member of] IAi respectively.

1.1.3 Partially ordered set

Let A be a set of elements a, b, c, d, ... of a certain nature. Let us introduce between certain pairs (a, b) of elements of A the relation a ≤ b with the properties:

(i) If a ≤ b and b ≤ c, then a ≤ c] (transitivity)

(ii) a ≤ a (reflexivity)

(iii) If a ≤ b and b ≤ a then a = b

Such a set A is said to be partially ordered by ≤ and a and b, satisfying a ≤ b and b [less than or equal to] a are said to be congruent. A set A is said to be totally ordered if for each pair of its elements a, b, a ≤ b or b ≤ a.

A subset B of a partially ordered set A is said to be bounded above if there is an element b such that y ≤ b for all y [member of] B, the element b is called an upper bound of B. The smallest of all upper bounds of B is called the least upper bound (l.u.b.) or supremum of B. The terms bounded below and greatest lower bound (g.l.b.) or infimum can be analogously defined. Finally, an element x0 [member of] A is said to be maximal if there exists in A no element xx0 satisfying the relation x0 ≤ x. The natural numbers are totally ordered but the branches of a tree are not. We next state a highly important lemma known as Zorn's lemma.

1.1.4 Zorn's lemma

Let X be a partially ordered set such that every totally ordered subset of X has an upper bound in X. Then X contains a maximal element.

Although the above statement is called a lemma it is actually an axiom.

1.1.5 Zermelo's theorem

Every set can be well ordered by introducing certain order relations.

The proof of Zermelo's theorem rests upon Zermelo's axiom of arbitrary choice, which is as follows:

If one system of nonempty, pair-wise disjoint sets is given, then there is a new set possessing exactly one element in common with each of the sets of the system.

Zorn's Lemma, Zermelo's Axiom of Choice and well ordering theorem are equivalent.

1.2 Function, Mapping

Given two nonempty sets X and Y, the Cartesian product of X and Y, denoted by X x Y is the set of all ordered pairs (x, y) such that x [member of] X and y [member of] Y.

Thus X × Y = {([x, y) : x [member of] X, y [member of] Y}.

1.2.1 Example

Let X = {a, b, c} and let Y = {d, e}. Then, X x Y = {(a, d]), (b, d), (c, d), (a, e), (b, e),(c, e)}.

It may be noted that the Cartesian product of two countable sets is countable.

1.2.2 Function

Let X and Y be two nonempty sets. A function from X to Y is a subset of X x Y with the property that no two members of f have the same first coordinate. Thus (x, y) [member of] f and (x, z) [member of] f imply that y = z. The domain of a function f from X to Y is the subset of X that consists of all first coordinates of members of f. Thus x is in the domain of f if and only if (x, y) [member of] f for some y [member of] Y.

The range of f is the subset of Y that consists of all second coordinates of members of f. Thus y is in the range of f if and only if (x, y) [member of] f for some x [member of] X. If f is a function and x is a point in the domain of f then f(x) is the second coordinate of the unique member of f whose first coordinate is x.

Thus y = f(x) if and only if (x, y) [member of] f. This point f(x) is called the image of x under f.

1.2.3 Mappings: into, onto (surjective), one-to-one (injective) and bijective

A function f is said to be a mapping of X into Y if the domain of f is X and the range of f is a subset of Y. A function f is said to be a mapping of X onto Y (surjective) if the domain of f is X and the range of f is Y.

The fact that f is a mapping of X onto Y is denoted by f: X [??] Y. A function f from X to Y is said to be one-to-one (injective) if distinct points in X have distinct images under f in Y. Thus f is one-to-one if and only if (x1, y) [member of] f and (x2, y) [member of] f imply that x1 = x2. A function from X to Y is said to be bijective if it is both injective and surjective.

1.2.4 Example

Let X = {a, b, c} and let Y = {d, e}. Consider the following subsets of X × Y:

F = {(a, b), (b, c), (c, d), (a, c)}, G = {(a, d), (b, d), (c, d)},

H = {(a, d), (b, e), (c, e)}, f = {(a, c), (b, d)}

The set F is not a function from X to Y because (a, d) and (a, e) are distinct members of F that have the same first coordinate. The domain of both G and H is X and the domain of φ is (a, b).

1.3 Linear Space

A nonempty set is said to be a space if the set is closed with respect to certain operations defined on it. It is apparent that the elements of some sets (i.e., set of finite matrices, set of functions, set of number sequences) are closed with respect to addition and multiplication by a scalar. Such sets have given rise to a space called linear space.

Definition. Let E be a set of elements of a certain nature satisfying the following axioms:

(i) E is an additive abelian group. This means that if x and y [member of] E, then their sum x + y also belongs to the same set E, where the operation of addition satisfies the following axioms:

(a) x + y = y + x (commutativity);

(b) x + (y + z) = (x + y) + z (associativity);

(c) There exists a uniquely defined element 0, such that x + θ = x for any x in E;

(d) For every element x [member of] E there exists a unique element (-x) of the same space, such that x + (-x) = θ.

(e) The element θ is said to be the null element or zero element of E and the element -x is called the inverse element of x.

(ii) A scalar multiplication is said to be defined if for every x [member of] E, for any scalar λ (real or complex) the element λx [member of] E and the following conditions are satisfied:

(a) λ(μx) = λ(μx) (associativity)

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) 1 x x = x

The set E satisfying the axioms (i) and (ii) is called a linear or vector space. This is said to be a real or complex space depending on whether the set of multipliers are real or complex.

1.3.1 Examples

(i) Real line R

The set of all real numbers for which the ordinary additions and multiplications are taken as linear operations, is a real linear space R.

(ii) The Euclidean space Rn, unitary space Cn, and complex plane C

Let X be the set of all ordered n-tuples of real numbers. If x = ([xi]1, [xi]2, ..., [xi]n and (ν1, ν2, ..., νn), we define the operations of addition and scalar multiplication as x + y [xi]1 + ν1 + [xi]2 + ν2, ..., [xi]n + νn) and λx = (λ[xi]1, (λ[xi]2, ..., (λ[xi]n). In the above equations, λ is a real scalar. The above linear space is called the real n-dimensional space and denoted by Rn. The set of all ordered n-tuples of complex numbers, Cn, is a linear space with the operations of additions and scalar multiplication defined as above. The complex plane C is a linear space with addition and multiplication of complex numbers taken as the linear operations over R (or C).

(Continues...)

Excerpted from A First Course in Functional Analysis by Rabindranath Sen. Copyright © 2013 Rabindranath Sen. Excerpted by permission of Wimbledon Publishing Company.
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