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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