Sequences and Series of Functions in Real Analysis
SEQUENCES AND SERIES OF FUNCTIONS
Prof. Mini Thomas
Department of Mathematics
CONTENT
INTRODUCTION
POINTWISE CONVERGENCE AND UNIFORM
CONVERGENCE OF SEQUENCES OF FUNCTIONS
CAUCHY
ʹ
S CRITERION FOR UNIFORM CONVERGENCE
THEOREMS
UNIFORM CONVERGENCE AND CONTINUITY
DEFINITION
CONCLUSION
REFERENCES
INTRODUCTION
In Mathematics , real analysis is the branch of
mathematical analysis that studies the behavior of real
numbers ,sequences and series of real numbers and
real valued functions .Some particular properties of
real –valued sequences and functions that real analysis
studies include convergence , limits , continuity ,
smoothness , differentiability and integrability .
POINTWISE CONVERGENCE AND UNIFORM
CONVERGENCE OF SEQUENCES OF FUNCTIONS
Definition ( Pointwise convergence )
Suppose {fn
} be a sequence of functions defined on a set E and
suppose that the sequence of numbers {fn
₍ₓ₎} converges for every
x
ϵ
E we can define a function f by
f
₍ₓ₎
= lim
n→∞
f
n
₍ₓ₎ , x
ϵ
E
And we say that {fn
} → f pointwise on E or simply , {fn
}
converges on E and that f is the limit or the limit function of
{fn
}.
⦁
Definition (Uniform convergence)
A sequence of functions {fn
} is said to converge uniformly on E
to a function if for every ɛ˃ 0 , their exist N such that
n ≥N implies │f
n
₍ₓ₎−f₍ₓ₎│≤ɛ , for all x
ϵ
E
CAUCHY ‘S CRITERION FOR
UNIFORM CONVERGENCE
THEOREM : The sequence of functions {fn
} defined on
E , converges uniformly on E iff for every ɛ˃0 their exist
an integer N such that
m,n ≥N , x
ϵ
E implies │f
n
₍ₓ₎─ f
m
₍ₓ₎│≤ɛ
THEOREMS
THEOREM 1: Suppose lim
n→∞
f
n
₍ₓ₎ = f₍ₓ₎ , x
ϵ
E put
M
n
=
x
ϵ
E
sup│f
n
₍ₓ₎─f₍ₓ₎│ then f
n
→f uniformly on E iff
M
n
→0 as n→∞
THEOREM 2 (WEIERSTRASS M-TEST FOR UNIFORM
CONVERGENCE OF SERIES ) :
Suppose {fn
} is a sequence of functions defined on E and
suppose │f
n
₍ₓ₎│≤M
n
, for all x
ϵ
E and n= 1,2,3,…… then
∑f
n
converges uniformly on E if ∑M
n
converges
UNIFORM CONVERGENCE AND
CONTINUITY
THEOREM
Suppose f
n
converges to f uniformly on a set E in a metric
space . Let x be a limit point of E and suppose that
lim
t→x
f
n
₍t₎= A
n
,n=1,2,3,. then {An
} converges and
lim
t→x
f₍t₎ = lim
n→∞
A
n
THEOREM : If {fn
} is a
sequence of continuous
functions on E and if f
n
converges to f uniformly on
E then f is continuous on E
THEOREM : suppose k is
compact
a)
{fn
} is a sequence of
continuous functions on
k
b)
{fn
} converges pointwise
to a continuous function f
on k
c)
f
n
₍ₓ₎ ≥ f
n+1
₍ₓ₎ , for every
x
ϵ
k , n =1,2,… then f
n
converges to f uniformly
on k
DEFINITION
Let X be a metric space then the set of all complex valued
continues bounded functions with domain x is denoted
by
Ҫ₍
ₓ₎ . If f is a continuous function on a compact
metric space then f is bounded . So if x is compact
Ҫ₍
ₓ₎
consists of complex valued continuous functions on X .
Let f
ϵ
Ҫ₍
ₓ₎ then define ǁ f ǁ =
x
ϵ
X
sup │f₍ₓ₎│, then ǁ ǁ is a
norm on
Ҫ₍
ₓ₎ and is called supremum norm
CONCLUSION
The difference between the uniform convergence and
pointwise convergence is that if {fn
} converges
uniformly on E which is possible for each
ԑ>0
to find
one integer N which will do for all x
ϵ
E so in case of
uniform convergence
N depends only on ԑ
and in case
of pointwise convergence
N depends on
ϵ
and x
REFERENCES
WALTER RUDIN, Principles Mathematical analysis (Third
edition) , McGraw Hill Book Company , International
Editions.
Chapter 7 Section 7.1- 7.18
Real analysis delves into the study of real numbers, sequences, series, and functions, exploring properties such as convergence, limits, continuity, differentiability, and integrability. This field scrutinizes the behavior of real-valued functions and their convergence types, including pointwise and uniform convergence, with criteria like Cauchy's criterion. Theorems such as the Weierstrass M-test for uniform convergence offer valuable insights, and the relationship between uniform convergence and continuity provides a deeper understanding of mathematical concepts in a metric space.
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Presentation Transcript
SEQUENCES AND SERIES OF FUNCTIONS Prof. Mini Thomas Department of Mathematics
CONTENT INTRODUCTION POINTWISE CONVERGENCE AND UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS CAUCHY S CRITERION FOR UNIFORM CONVERGENCE THEOREMS UNIFORM CONVERGENCE AND CONTINUITY DEFINITION CONCLUSION REFERENCES
INTRODUCTION In Mathematics , real analysis is the branch of mathematical analysis that studies the behavior of real numbers ,sequences and series of real numbers and real valued functions .Some particular properties of real valued sequences and functions that real analysis studies include convergence , limits , continuity , smoothness , differentiability and integrability .
POINTWISE CONVERGENCE AND UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS Definition ( Pointwise convergence ) Suppose {fn} be a sequence of functions defined on a set E and suppose that the sequence of numbers {fn } converges for every x E we can define a function f by f = limn fn , x E And we say that {fn} f pointwise on E or simply , {fn} converges on E and that f is the limit or the limit function of {fn}. Definition (Uniform convergence) A sequence of functions {fn} is said to converge uniformly on E to a function if for every 0 , their exist N such that n N implies fn f , for all x E
CAUCHY S CRITERION FOR UNIFORM CONVERGENCE THEOREM : The sequence of functions {fn} defined on E , converges uniformly on E iff for every 0 their exist an integer N such that m,n N , x E implies fn fm
THEOREMS THEOREM 1: Suppose limn fn = f , x E put Mn = x Esup fn f then fn f uniformly on E iff Mn 0 as n THEOREM 2 (WEIERSTRASS M-TEST FOR UNIFORM CONVERGENCE OF SERIES ) : Suppose {fn} is a sequence of functions defined on E and suppose fn Mn , for all x E and n= 1,2,3, then fnconverges uniformly on E if Mn converges
UNIFORM CONVERGENCE AND CONTINUITY THEOREM Suppose fn converges to f uniformly on a set E in a metric space . Let x be a limit point of E and suppose that limt xfn t = An ,n=1,2,3,. then {An} converges and limt xf t = limn An
THEOREM : If {fn} is a sequence of continuous functions on E and if fn converges to f uniformly on E then f is continuous on E THEOREM : suppose k is compact a) {fn} is a sequence of continuous functions on k b) {fn} converges pointwise to a continuous function f on k c) fn fn+1 , for every x k , n =1,2, then fn converges to f uniformly on k
DEFINITION Let X be a metric space then the set of all complex valued continues bounded functions with domain x is denoted by . If f is a continuous function on a compact metric space then f is bounded . So if x is compact consists of complex valued continuous functions on X . Let f then define f = x Xsup f , then is a norm on and is called supremum norm
CONCLUSION The difference between the uniform convergence and pointwise convergence is that if {fn} converges uniformly on E which is possible for each >0 to find one integer N which will do for all x E so in case of uniform convergence N depends only on and in case of pointwise convergence N depends on and x
REFERENCES WALTER RUDIN, Principles Mathematical analysis (Third edition) , McGraw Hill Book Company , International Editions. Chapter 7 Section 7.1- 7.18
