Mathematics ( Paper-I ) Metric Spaces & Complex Analysis Book B.Sc 6th Semester U.P

Part A

Unit 1: Basic Concepts of Metric Space

¼bdkbZ 1& nwjhd Lkef"V dh ewy vo/kkj.kk,¡½

1.1.

Metric Spaces

ehfVªd Lisl ¼nwjhd lef"V½

13

1.1.1.

Definition

ifjHkk"kk

13

1.1.2.

Usual Metric

çkf;d nwjhd

17

1.1.3.

Pseudo-Metric

Nn~e &nwjhd

21

1.1.4.

Discrete Metric

fofoDr nwjhd

22

1.1.5.

Norm Vector Space

u‚eZ lfn’k Lkef"V

24

1.1.6.

Quasi Metric Space

v/kZ nwjhd Lkef"V

24

1.1.7.

Bounded And Unbounded Metric Space

ifjc) rFkk vifjc) nwjhd Lkef"V

25

1.2.

Sequences in Metric Spaces

nwjhd Lkef"V;ksa esa vuqØe

27

1.2.1.

Introduction

ifjp;

28

1.2.2.

Convergence of the Sequence

vuqØe dh vfHklkfjrk

28

1.2.3.

Cauchy Sequences

d‚ph vuqØe

30

1.2.4.

Properties of Sequence

vuqØe ds xq.k

30

1.2.5.

Theorems

çes;

30

1.2.6.

Complete/ Completeness Metric Space

iw.kZ@iw.kZrk nwjhd Lkef"V

36

1.3.

Multiple Choice Questions

cgq fodYih; ç'u

39

Unit 2: Topology of Metric Spaces

¼bdkbZ 2& nwjhd Lkef"V dh Vksiksy‚th½

2.1.

Balls or Spheres

xsansa ;k xksyd

42

2.1.1.

Open And Closed Ball

foo`r vkSj lao`r xksyd

42

2.1.2.

Neighborhood

çfros’k ;k lkehI;

43

2.2.

Open Set

foo`r leqPp;

46

2.2.1.

Properties of Open Sets

foo`r leqPp; ds çxq.k

46

2.2.2.

Interior, Exterior And Frontier Point

vkarfjd]  cká  vkSj lhekar fcanq

50

2.2.3.

Limit And Boundary Point of A Set

,d leqPp; dh lhek vkSj lhek fcanq

53

2.2.4.

Derived Set

O;qRiUu leqPp;

55

2.3.

Closed Set

lao`r leqPp;

59

2.3.1.

Introduction

ifjp;

59

2.3.2.

Properties of Closed Set

lao`r leqPp; ds çxq.k

59

2.3.3.

Relationship Between The Open And Closed Sets

foo`r vkSj lao`r leqPp; ds chp laca/k

63

2.3.4.

Closure of A Set

leqPp; dk laojd

65

2.3.5.

Relation Between Interior And Closure of A Set

fdlh leqPp; ds vkarfjd vkSj laojd ds chp laca/k

67

2.4.

Product of Matric Spaces

nwjhd lef"V dk xq.ku

67

2.5.

Diameter of A Set

,d leqPp; dk O;kl

68

2.5.1.

Introduction

ifjp;

68

2.5.2.

Distance Between Two Sets

nks leqPp;ksa ds chp nwjh

70

2.6.

Cantor’s Theorem

dSaVj dk çes;

71

2.6.1.

Introduction

ifjp;

71

2.6.2.

Cantor Intersection Theorem

dSaVj ÁfrPNsnu çes;

71

2.7.

Subspaces

milef"V

73

2.8.

Dense Set

l?ku leqPp;

74

2.9.

Multiple Choice Questions

cgq fodYih; ç'u

79

2.10.

Exercise

vH;kl

80

Unit 3: Continuity & Uniform Continuity in Metric Spaces

¼bdkbZ 3& nwjhd lef"V esa lkarR;rk vkSj ,dleku lkarR;rk½

3.1.

Continuity In Metric Spaces

nwjhd lef"V esa lkarR;rk

82

3.1.1.

Limit of Functions

Qyuksa dh lhek

82

3.1.2.

Continuity of Function

Qyu dh lkarR;rk

82

3.1.3.

Mapping

çfrfp=.k

86

3.1.4.

Continuous Mapping

larr çfrfp=.k

86

3.1.4.1.

Locally Continuous Mapping

LFkkuh; :i ls larr çfrfp=.k

87

3.1.4.2.

Composition of Continuous Mapping

larr çfrfp=.k dh lajpuk

88

3.1.5.

Topological Mapping

Vksiksy‚ftdy çfrfp=.k

88

3.1.6.

Topological Equivalence

Vksiksy‚ftdy rqY;rk

88

3.1.7.

Sequential Criterion

vuqØfed ekunaM

89

3.1.8.

Characterizations of Continuity

lkarR;rk ds y{k.k

90

3.2.

Uniform Continuity

,dleku lkarR;rk

92

3.3.

Homeomorphism

le:irk

95

3.3.1.

Introduction

ifjp;

95

3.3.2.

Homeomorphic Spaces

gksfe;ksekWfQZd lef"V

96

3.4.

Fixed Point Theorems

fuf'pr fcanq çes;

97

3.4.1.