Sanjay Gupta, Assistant Professor, Post Graduate Department of .... POINTWISE AND UNIFORM CONVERGENCE OF SEQUENCES OF FU
SANJAY GUPTA, DEV SAMAJ COLLEGE FOR WOMEN, FEROZEPUR
Pointwise and uniform convergence of sequences of functions Sanjay Gupta, Assistant Professor, Post Graduate Department of Mathematics, Dev Samaj College For Women, Ferozepur City, Email –
[email protected]
INTRODUCTION Here, we shall define and study the convergence of sequences of functions. There are many different ways to define the convergence of a sequence of functions and different definitions lead to inequivalent types of convergence. We consider here two basic types : Pointwise and Uniform Convergence. DEFINITIONS AND EXAMPLES 1. Sequence of real valued functions. Let be a real valued function defined on an interval ⊆ for each ∈ . Then { , , , … , , … } is called a sequence of real valued functions on . It is denoted by { } or 〈 〉. Example 1. If {
( ), ( ),
( ), … } = { ,
( )=
,0 ≤
≤ 1,
, … } is a sequence of real valued functions on [0,1].
,
sanjay gupta
then
be a real valued function defined by
Example 2. If then
{
be a real valued function defined by ( ), ( ),
( ), … } =
,
,
( )=
,0 ≤
≤ 1,
, … is a sequence of real valued functions on [0,1].
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SANJAY GUPTA, DEV SAMAJ COLLEGE FOR WOMEN, FEROZEPUR 2. Sequence of real numbers. Let { } be a sequence of real valued functions defined on an interval , then for ∈ , { ( )} = { ( ), ( ), ( ), … , ( ), … } is a sequence of real numbers. Example 3. Let {
} be a sequence of real valued functions defined by
(i)
=
,
,
,…,
,… =
,
sanjay gupta
is a sequence of real numbers corresponding to the point
(ii)
{
(0) } = {
(0),
(0), (0), … ,
{
(1) } = {
(1),
(1), (1), … ,
,…,
≤ 1, then
,…
∈ [0,1].
=
= 0 ∈ [0,1].
(1), … } = { 1,1,1, … ,1, … }
is a sequence of real numbers corresponding to the point Thus to each
,
,0 ≤
(0), … } = { 0,0,0, … ,0, … }
is a sequence of real numbers corresponding to the point (ii)
( )=
= 1 ∈ [0,1].
∈ , we have a sequence of real numbers.
3. Convergence of sequence of real numbers. A sequence { } of real numbers is said to converge to a real number if for given > 0 ( however small ), there exist a positive integer ( depending upon ) such that |
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