Book contents
- Frontmatter
- Preface
- Contents
- Introduction
- I Preliminaries
- II Normed Linear Spaces
- III Hilbert Space
- IV Linear Operators
- V Linear Functionals
- VI Space of Bounded Linear Functionals
- VII Closed Graph Theorem and Its Consequences
- VIII Compact Operators on Normed Linear Spaces
- IX Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces
- X Measure and Integration in Lp Spaces
- XI Unbounded Linear Operators
- XII The Hahn-Banach Theorem and Optimization Problems
- XIII Variational Problems
- XIV The Wavelet Analysis
- XV Dynamical Systems
- List of Symbols
- Bibliography
- Index
- Frontmatter
- Preface
- Contents
- Introduction
- I Preliminaries
- II Normed Linear Spaces
- III Hilbert Space
- IV Linear Operators
- V Linear Functionals
- VI Space of Bounded Linear Functionals
- VII Closed Graph Theorem and Its Consequences
- VIII Compact Operators on Normed Linear Spaces
- IX Elements of Spectral Theory of Self-Adjoint Operators in Hilbert Spaces
- X Measure and Integration in Lp Spaces
- XI Unbounded Linear Operators
- XII The Hahn-Banach Theorem and Optimization Problems
- XIII Variational Problems
- XIV The Wavelet Analysis
- XV Dynamical Systems
- List of Symbols
- Bibliography
- Index
Summary
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.
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 ∈ 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}.
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- Information
- A First Course in Functional AnalysisTheory and Applications, pp. 1 - 57Publisher: Anthem PressPrint publication year: 2013