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    A First Course in Functional Analysis
    A First Course in Functional Analysis
    A First Course in Functional Analysis
    Ebook161 pages57 minutesDover Books on Mathematics

    A First Course in Functional Analysis

    By Martin Davis

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    This introduction to functional analysis is based on the lecture notes of Martin Davis, a distinguished professor of mathematics. The treatment demonstrates the essential unity of mathematics without assuming more background than can be expected of advanced undergraduates and graduate students majoring in mathematics.
    A self-contained exposition of Gelfand's proof of Wiener's theorem, this volume explores set theoretic preliminaries, normed linear spaces and algebras, functions on Banach spaces, homomorphisms on normed linear spaces, and analytic functions into a Banach space. Numerous problems appear throughout the book.
    LanguageEnglish
    PublisherDover Publications
    Release dateMay 27, 2013
    ISBN9780486315812
    A First Course in Functional Analysis

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      Book preview

      A First Course in Functional Analysis - Martin Davis

      CHAPTER 1

      Set Theoretic Preliminaries

      1. Sets and Members

      In this section the terms class, set, collection, totality and family will be used synonomously. The symbol, ∈(epsilon), will be used to denote membership in a class. The symbol, ∉, will denote non-membership in a class, i.e.,

      x ∈ C means x is a member of set C

      x ∉ C means x is not a member of set C

      Example: If C is the class of all even numbers, then 2 belongs to C, (2 ∈ C), 4 belongs to C, (4 ∈ C), and 3 does not belong to C, (3 ∉ C).

      If a set is finite, then it can be described by listing its members. For example:

      Let C be the set {1, 3, 5} then 1 ∈ C, 3 ∈ C, 5 ∈ C and all other elements are not members of the set C. The description of an infinite set, however, is not so simple. An infinite set can not be described by simply listing its elements. Hence, we must have recourse to defining the set by a characteristic property. The set can then be described as the set of all elements which possess the property in question. If the property is, say, P(x), then the set will be written {x | P(x)}, thus to write C = {x | P(x)}, is to say that C is the set of all elements x, such that, P(x) is true. For example,

      {a, b} = {x|x = a or x = b}

      A set may have only one element: {a} = {x|x = a}.

      Definition 1.1. The union of set A with set B, A ∪ B is the set of all those elements which belong either to set A or to set B or to both. Symbolically,

      A ∪ B = {x|x ∈ A or x ∈ B}

      Definition 1.2. The intersection of two sets, A ∩ B, is defined to be the set of all those elements which are common to both set A and set B. Symbolically,

      A ∩ B = {x|x ∈ A and x ∈ B}

      Definition 1.3. The difference between two sets, denoted by A − B is defined to be the set

      A − B = {x|x ∈ A and x ∉ B}

      We shall use the symbols:

      ⇒ to mean implies

      ⇔ to mean if and only if and

      ∃ to mean there is a

      Actually a certain amount of care is necessary in using the operation: {x| ... ...}. E.g., let P(x) be the property: x ∉ x. (Thus, we could plausibly claim as an x for which P(x) is true, say, the class of even numbers, whereas the class of, say, non-automobiles, or the class of entities definable in less than 100 English words, could be claimed as x's making P(x)false.) Suppose we can form:

      Applying this to x = A, yields

      A ∈ A ⇔ A ∉ A

      a contradiction. This is Russell's paradox.

      In an-axiomatic treatment of set theory (cf. Kelley, General Topology, Appendix) suitable restrictions have to be placed on the operation: {x| ... ...} in order to prevent the appearance of the Russell paradox or similar paradoxes. In these lectures we shall use this operation whenever necessary. However, all of our uses could be justified in axiomatic set theory.

      The symbol = of equality always will mean absolute identity. In particular, for sets A, B, the assertion A = B means that the sets A and B have the same members.

      Definition 1.4. The set A is a subset of a set B, written A ⊂ B, if each element of A is also an element of B, that is if x ∈ A ⇒ x ∈ B.

      From this definition it follows that A ⊂ A.

      Definition 1.5. The empty set, represented by the symbol is the set which contains no elements.

      For example:

      = {x|x ≠ x}

      = {x|x = 0 and x = 1}

      The empty set is a subset of any set, i.e. ⊂ A.

      Definition 1.6. 2A is the class of all sets B, such that B is a subset of A, i.e.

      2A = {B|B ⊂ A}

      For example, consider the set A, where

      A = {L, M, N}

      Then the elements of 2A are the empty set , the sets containing only one element, {L}, {M}, {N}, the sets containing two elements, {L, N}, {L, M}, {M, N}, and finally the single set containing all three elements {L, M, N}. Note that A contains 3 elements and that 2A contains 8 = 2³ elements.

      Definition 1.7. The ordered pair of two elements is defined as (a, b)= {{a}, {a, b}}.

      It can easily be shown that (a, b) = (a′, b′) ⇒ a = a′ and b = b′. (Cf. Problem 1.) It is this which is the crucial property of the ordered pair. Any other construct with this property could be used instead.

      Note that (a, b) is quite different from {a, b}. For, {a, b} is always equal to {b, a}.

      Definition 1.8. The Cartesian product of sets A and B, written, A × B, is the set of all ordered pairs (a, b), such that, a belongs to the set A, and b belongs to the set B, i.e.,

      A × B = {(a, b)|a ∈ A and b ∈ B}

      For example, if:

      then the Cartesian product is the set

      Note that there are 6 (= 3 × 2) elements in the set.

      2. Relations

      Definition 2.1. A relation between sets A and B is a set C such that C ⊂ A × B.

      Examples of such a C are {(L, P), (L, Q}, {(M, P), (M, Q)}, and {(N, P), (N, Q} as taken from the above example.

      Definition 2.2. A relation on A is a relation between A and A.

      Definition 2.3. A mapping, transformation or function α from A into B is a relation between A and B, such that, for each x ∈ A, there is exactly one y ∈ B such that (x, y) ∈ α.

      We write: α(x) = y to mean (x, y) ∈ α.

      For example, let A be a set of positive integers and C be the set of ordered pairs {a, b) such that a − b is even, i.e. C = {{a, b) | a − b is even}. Then,

      (1, 3) ∈ C

      (2, 4) ∈ C

      (1, 6) ∉ C

      Specifically, (a, b) ∈ C ⇔ a ≡ b mod 2. C is a relation on A.

      Example: Let A be the set of human beings, and let α be the set of pairs (x, y) where x ∈ A, y ∈ A and y is the father of x. Then, α is a relation on A and is also a function from A into A. However (cf. Def. 2.4 below), it is not a function from A onto A.

      Definition 2.4. A function α is called a mapping from A onto B, if for each y ∈ B there is at least one x ∈ A such that (x, y) ∈ α.

      Definition

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