[PDF] STUDY MATERIAL FOR B.SC. MATHEMATICS REAL ANALYSIS II

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STUDY MATERIAL FOR B.SC. MATHEMATICS

REAL ANALYSIS II

SEMESTER - V, ACADEMIC YEAR 2020 - 21

Page 1 of 47

UNIT CONTENT PAGE Nr

I METRIC SPACES 02

II CLOSED SETS 12

III CONTINUOUS FUNCTIONS ON METRIC SPACES 26

IV CONNECTEDNESS AND COMPACTNESS 34

V RIEMANN INTEGRAL 42

STUDY MATERIAL FOR B.SC. MATHEMATICS

REAL ANALYSIS II

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Page 2 of 47

UNIT - I

METRIC SPACES

Introduction

A Metric Space is a set equipped with a distance function, also called a metric, which enables us to measure the distance between two elements in the set.

1.1 Definition andExamples

Definition: A Metric Space is a non empty set M together with a functionࢊ׷ satisfying the following conditions. ݀ is called a metric ordistance function on ܯ by ܯ

Example 1.

Let ࡾ be the set of all real numbers. Define a function ݀ܯ ׷ ൈ ܯ ՜ ܴ

Proof.

Let ݔ ǡݕ א

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Let ݔ ǡݕ ǡݖ א

Hence ݀ is a metric on ࡾ.

Note. When ܴ

metric.

Example 2

Let ࡹ be any non-empty set. Define a function ࢊ׷

Proof.

Let ݔ ǡݕ ܯ א

Let ݔ ǡݕ ǡݖ ܯ א

Case (i) Suppose ݔ ൌ ݕ ൌ ݖ.

Case (ii) Suppose ݔ ൌ ݕ and ݖ distinct.

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Case (iii) Suppose ݔ ൌ ݖ and ݕ distinct. Then, Case (iv) Suppose ݕ ൌ ݖ and ݔ distinct. Then,

In all the cases,

Hence ݀ is a metric on ܯ

1.2.Open Sets in aMetricSpace

Examples:

radiusݎ.

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Examples:

For,

1. Every subset of a discrete metric space ܯ

Let ܣ be a subset of ܯ

Otherwise, let ݔ ܣ א

2. Set of all rational numbers ࡽ is not open in ࡾ. For,

Let ݔא

numbers.

Theorem 1.1

Proof.

Now,

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Theorem1.2

In any metric space ܯ

Proof.

We have to prove ܣ ൌ ܣ׫

Then ݔ ܣא௜ for some ܫ א

Hence ܣ is open in ܯ

Theorem 1.3

In any metric space ܯ

Proof:

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Then,ݔ ܣא

Clearly,ݎ ൐ - and

ܣ ׵is open in ܯ

Theorem 1.4

as union of open balls.

Proof :Suppose that ܣ is open in ܯ

Thus ܣ

Conversely, assume that ܣ

open balls are open and union of open sets is open, ܣ

1.2 Interior of aset

Example: In ࡾwith usual metric, let ܣ

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Note:

Theorem1.5

Proof.

Let ܩ ൌ ׫

Then ݔ is an interior point of ܣ

Let ܩא

׵ ݔis an interior point of ܣ

Theorem1.6

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Proof.

For,

Inࡾwith usual metric,

1.2.Subspace

Definition:

Theorem1.8Let ܯ be a metric space and ܯଵ a subspace of ܯ. Let ܯكܣଵ. Then ܣ

ܯଵ if and only if ܣଵ с A ൘ ܯଵ where ܣ

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Proof:

Let ܣ

Then, = A ܯځ Conversely, let ܣൌܯתܩଵ where ܩ is open in ܯ

We shall prove that ܣ

Let ݔܣא

Then ݔܽ ܣא݊݀ ݔܯא

Since ܣ is open in ܯ, there exists an open ball B(x,r) such that B(x,r)ك

ܣ׵ଵis open in ܯ

1.2.Bounded Sets in a Metric space.

Let ܣ be any finite subset of ܯ

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Example:Any subset ܣ of a discrete metric space ܯ

Example:Inܴ

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UNIT - II

CLOSED SETS

2.1.ClosedSets

Definition: A subset ܣ of a metric space ܯ is said to be closed in ܯ

Examples

1. Any subset ܣ of a discrete metric space ܯ is closed since ܣ௖is open as every subset of ܯ

isopen.

Theorem 2.1.

In any metric space ܯ

Proof:

Let ࢞א

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Let ࢟א

Hence ࢟א

Theorem 2.2

In any metric space ܯ

Proof:

We have to prove ܣځ௜௜א

(byDeMorgans law)

Since ܣ௜ is closed ܣ

Hence ܣڂ௜௖௜א

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ܣځ׵௜௜אூ is closed in ܯ

Theorem 2.3

Let ܯଵ be a subspace of a metric space ܯ. Let ܨଵܯكଵ. Then ܨଵ is closed in ܯ

ܨଵൌܯתܨଵ where ܨ is a closed set in ܯ

Proof.

Suppose that ܨis closed in ܯ

Then ܯଵ Ȃ ܨଵ is open inܯ

ܯ׵ଵȂ ܨଵൌ ܣ௖ܯתଵwhereܣ is open in ܯ

Now, ܨଵൌܯתܣ

Since ܣ is open in ܣ, ܯ௖ is closed in ܯ Thus, ܨଵൌ ܯתܨଵwhere ܨൌܣ௖ is closed in ܯ Conversely, assume that ܨଵൌ ܯתܨଵ where ܨ is closed in ܯ Since ܨ is closed in ܨ, ܯ௖is open in ܯ

ܨ׵௖ܯתଵ is open in ܯ

Now, ܯଵȂ ܨଵൌ ܨ௖ܯתଵ which is open inܯ

ܨ׵ଵis closed inܯ

Proof of the converse is similar.

2.1.Closure.

to be the intersection of all closed sets which contain ܣ Note (1) Since intersection of closed sets is closed, ܣ (2) ܣଵis the smallest closed set containingܣ (3) A is closed ֞ A =ܣ

Theorem 2.4:

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Proof:

(i)let࡭ك But ܣҧ is the smallest closed set containing ܣ (ii)we have A ك A ׫

2.1.Limit Point

if every open ball with centre ࢞ contains atleast one point of A differ from ݔ.

Theorem 2.4

ball with center ݔ contains infinite number of points of ܣ

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Proof:

Let ࢞ be a limit point of ࡭.

infinite number of points of ࡭.

The converse is obvious.

Corollary 1: Any finite subset of a metric space has no limit points.

Theorem 2.5

Let ࡹ be a metric space and ࡭كࡹ. then ࡭ഥൌ࡭׫

Suppose࢞ב

If ࢞א࡭clearly ࢞א࡭׫

Suppose ࢞ב࡭. We claim that ࢞א

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But ࡭ഥ is the smallest closed set containing A. but࢞א࡭ഥ and ࢞ב

Hence ࢞א

Hence ׵ ࡭׫

Corollary 1:࡭ is closed iff ࡭ contains all its limit points.

Proof:࡭ is closed ֞

Proof: let ࢞א࡭ഥ,then࢞א࡭׫

We have to prove that, ࢞א

If ࢞א࡭trivially ࢞א

Let ࢞ב

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Corollary 3:

Proof: Let ࢞א

Conversely suppose ࡳת

2.1.Dense sets

Definition:A subset ࡭ of a metric space ࡹ is said to be dense in ࡹor every where dense if ࡭ഥൌ

Definition: A metric space ࡹ is said to be separable if there exists a countable dense subset in ࡹ.

Note :

(1) Any countable metric space is separable. (2)Any uncountable discrete metric space is not separable.

Theorem 2.6:

Let ࡹ be a metric space and ࡭ك

(i) ࡭is dense in ࡹ. (ii) The only closed set which contains ࡭is ࡹ. (iii) The only open set disjoint from ࡭is ࣘ. (iv) ࡭ intersects every non empty open set. (v) ࡭intersects every open ball.

Proof:

(i)֜

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Suppose ࡭is dence in ࡹ.

Then ࡭ഥൌࡹ. -------------------------- (1)

Now, let ࡲك

Since ࡭ഥ is a closed set containing ࡭,we have ࡭ഥك

Hence ࡹك

Hence,The only closed set which contains ࡭ is ࡹ. (ii)֜

Suppose (iii) is not true.

Then there exists a non emptyopenset࡮ such that,࡮ת ׵࡮ࢉis z closed set and ࡮ࢉل Further, since ࡮്ࣘ we have ࡮ࢉ്ࡹ which is a contradiction to (ii).

Hence (ii))֜

Obviously, (iii)֜

(iv)֜ (iv) ֜

Let ࢞א

Then by corollary, ࢞א

׵ࡹك࡭ഥ. But trivially ࡭ഥك

2.2.Completeness

ࡹ. Let ࢞א

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such that ࢞࢔א

Theorem2.6:

૛ࢿ for all ࢔൒࢔૛.

Theorem2.7:

let ࡹ be a metric space and ࡭ك (vi) ࢞א

Proof:

Let ࢞א

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Then, ࢞א࡭׫

׵࢞א࡭or࢞א

If ࢞א

If ࢞א

࢔ for all ࢔. ࢿ for all ࢓ǡ࢔൒࢔૙.

Theorem 2.7:

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