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


THIS IS A SAMPLE | WE ARE PROFESSIONALS IN WRITING



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