Theory of Linear Operations

S. Banach

• Publisher: North Holland
• Year: 1987
• ISBN:
  9780080887203
• Topic: Computer Science
• Subtopic: Computer Science General
• Index: Computer Science

This classic work by the late Stefan Banach has been translated into English so as to reach a yet wider audience. It contains the basics of the algebra of operators, concentrating on the study of linear operators, which corresponds to that of the linear forms a 1 x 1 + a 2 x 2 +... + a n x n of algebra.The book gathers results concerning linear operators defined in general spaces of a certain kind, principally in Banach spaces, examples of which are: the space of continuous functions, that of the p th -power-summable functions, Hilbert space, etc. The general theorems are interpreted in various mathematical areas, such as group theory, differential equations, integral equations, equations with infinitely many unknowns, functions of a real variable, summation methods and orthogonal series.A new fifty-page section (``Some Aspects of the Present Theory of Banach Spaces'') complements this important monograph.

Chapter I

Groups

§1. Definition of G-spaces

Let a complete metric space E be given. Suppose that to each ordered pair (x,y) of elements of the space E there corresponds a unique element z of this space called the sum of x and y and which we will denote by the symbol x + y.

Suppose further that E is a group under this sum operation, i.e. that

I₁.

I₂. there exists in E a zero–element θ such that one has

I₃. to each element x of E there corresponds an element (which we will denote by -x) which satisfies the equation

It follows easily from these axioms that:

a) there exists only one zero–element θ in E,

b) one has (−x) + (x) = θ for each x ε E.

c) x + y = x + z implies y = z.

Suppose further that the following axioms are satisfied:

II₁.

II₂.

The complete metric spaces satisfying these axioms will be called G-spaces.

Remark. We will write x − y instead of x + (−y) and −x + y instead of (−x) + y.

§2. Properties of sub-groups

Let E be a G-space. For an element x ε E and a set H ⊆ E, we will denote by xH and Hx respectively the set of all elements y ε E such that y = x + z (z + x, respectively) where z ε H.

Clearly, one always has the identities

and the analogous identities for H₁x and H₂x.

It is easily shown that if H has any of the properties closed, open, nowhere dense, of category I, of category II or B-measurable then the set xH also has the same properties. If z is an interior point of H, x + z is an interior point of xH.

A non-empty set H ⊆ E is called a subgroup of E, when the conditions x ε H and y ε H imply x + y ε H and −x ε H. Clearly then also θ ε H.

A set is said to be connected when it cannot be expressed as the union of two non-empty disjoint relatively closed subsets of itself. If E is a connected set and H is a subset of E which is both open and closed, one has H = E, for otherwise the set E\H would also be non-empty and closed.

Theorem 1.

Every sub...

Table of contents

• Cover image
• Title page
• Table of Contents
• North-Holland Mathematical Library
• Copyright page
• Preface
• Introduction
• Chapter I: Groups
• Chapter II: General vector spaces
• Chapter III: F-spaces
• Chapter IV: Normed spaces
• Chapter V: Banach spaces
• Chapter VI: Compact operators
• Chapter VII: Biorthogonal sequences
• Chapter VIII: Linear functionals
• Chapter IX: Weakly convergent sequences
• Chapter X: Linear functional equations
• Chapter XI: Isometry, equivalence, isomorphism
• Chapter XII: Linear dimension
• Weak convergence in Banach spaces
• Remarks
• Index
• Some aspects of the present theory of Banach spaces
• Bibliography