COMPLETENESS IN METRIC SPACES | COMPLETE PROJECT WORK

ABSTRACT

A metric space (X, r) is said to be complete if every Cauchy sequence in x converges to a point of X. Otherwise such a space is said not to be complete that is incomplete. In this work, we discuss the concept of completeness in metric spaces and their basic consequences using a good number of instructive examples.

DEPARTMENT OF INDUSTRIAL MATHEMATICS AND APPLIED STATISTICS

FACULTY OF PHYSICAL SCIENCE

IN PARTIAL FULFILLMENT OF COURSE REQUIREMENT FOR THE AWARD OF BACHELOR OF SCIENCE (B. Sc.) IN MATHEMATICS AND COMPUTER

TABLE OF CONTENTS

Title page……………………………………………………….    i

Approved page………………………………………………….    ii

Dedication………………………………………………………    iii

Acknowledgements ……………………………………………..   iv

Abstract…………………………………………………………    v

Table of contents……………………………………………….    vi

CHAPTER ONE

Metric Spaces

1.1     Introduction……………………………………………..      1

1.2     Definitions………………………………………………      2

1.3     Some Example of metric Spaces……………………….      2

1.4     Minkowski’s inequality, Holder’s inequality,

Schwartz’ inequality…………………………………….     13

CHAPTER TWO

Geometry of metric Spaces

2.1     Open ball, Closed ball, sphere…………………………..     17

2.2     Open set, Closed set…………………………………….      22

2.3     Continuous functions, bounded set, distance

between sets……………………………………………..     29

2.4     The product of metric spaces…………………………….    31

CHAPTER THREE

Sequences in Metric Spaces

3.1     Definition some examples……………………………….    33

3.2     Convergence of a sequence, Limit……………………….    34

3.3     Cauchy sequence………………………………………….   37

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