Adam Bowers - An Introductory Course in Functional Analysis

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Springer Science+Business Media, LLC 2014

Adam Bowers and Nigel J. Kalton An Introductory Course in Functional Analysis Universitext 10.1007/978-1-4939-1945-1_1

1. Introduction

Adam Bowers 1

(1)

Department of Mathematics, University of California, San Diego, La Jolla, CA, USA

(2)

Department of Mathematics, University of Missouri, Columbia, Columbia, MO, USA

Adam Bowers (Corresponding author)

Email:

Nigel J. Kalton (deceased)

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Functional analysis is at its foundation the study of infinite-dimensional vector spaces. The goal of functional analysis is to generalize the well-known and very successful results of linear algebra on finite-dimensional vector spaces to the more complicated and subtle infinite-dimensional spaces. Of course, in infinite dimensions, certain issues (such as summability) become much more delicate, and accordingly additional structure is imposed. Perhaps the most natural structure imposed upon a vector space is that of a norm .

Definition 1.1

A normed space is a real or complex vector space X together with a real-valued function defined for all , called a norm , such that

• (N1) for all , and if and only if x =0,
• (N2) for all x and y in X , and
• (N3) for all and scalars .

Property (N1) is known as non-negativity or positive-definiteness , Property (N2) is called subadditivity or the triangle inequality , and Property (N3) is called homogeneity .

A norm on a vector space X is essentially a way of measuring the size of an element in X . If x is in (or ), the norm of x is given by the absolute value (or modulus) of x , which is denoted (in either case) by .

Definition 1.2

A metric space is a set X together with a map that satisfies the following properties:

• (M1) for all x and y in X , and if and only if x=y ,
• (M2) for all x , y , and z in X , and
• (M3) for all x and y in X .

The function d is said to be a metric on X . As was the case with a norm, (M1) is known as non-negativity and (M2) is called the triangle inequality . The final property, (M3), is called symmetry .

A metric is a measure of distance on the set X . Any normed space is metrizable ; that is, we can always introduce a metric on a normed space by

This metric measures the size of the displacement between x and y . While every norm determines a metric, not every metric space can have a norm. (We will see some examples of this in Sect..)

Definition 1.3

Let X be a metric space with metric d . A sequence in X is said to converge to a point x in X if for any there exists an such that whenever . In such a case, we say the sequence is convergent and write . A sequence in X is called a Cauchy sequence if for any there exists an such that whenever and .

It is well-known that a scalar-valued sequence converges if and only if it is a Cauchy sequence. This is not always true in infinite-dimensional normed spaces. A convergent sequence will always be a Cauchy sequence, but there may be Cauchy sequences that do not converge to an element of the normed space. Such a space is called incomplete , because we imagine it lacks certain desirable points. For this reason, we are generally interested in normed spaces that are complete ; that is, spaces in which every Cauchy sequence does converge. For these spaces we have a special name.

Definition 1.4

A normed space X is called a Banach space if it is a complete metric space in the metric given by for all

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