International Journal of Trend in Scientific
Research and Development (IJTSRD)
International Open Access Journal
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 1
Analyzation of Weak Convergence on L^p Space
R. Ranjitha
Research Scholar, Department of
Mathematics, Vivekanandha
College of Arts and Sciences for
Women, Tiruchengode,
Tamilnadu, India
S. Dickson
Assistant Professor, Department of
Mathematics, Vivekanandha
College for Women,
Tiruchengode, Tami
Tamilnadu, India
ABSTRACT
In this paper a study on duality, weak convergences and
weak sequential compactness on 𝐿 space. A new
generalized 𝐿 space is investigated from the weak
sequential convergence. Some more special types and
properties of 𝐿 are presented. Few relevant examples
are also included to justify the proposed notions.
Keywords: Dual space, Linear functional, Weak
Convergence, Compact
1. INTRODUCTION
The Lp spaces were introduced by FrigyesRiesz [9] as
part of a program to formulate for functional and
mappings defined on infinite dimensional spaces
appropriate versions of properties possessed by
functionals and mappings
ings defined on finite dimensional
spaces. In this paper, we study the weak convergence
and weak compactness on the so-called
called Lebe
Lebesgue
spaces (or Lp – spaces as they are usually called).
2. PRELIMINARIES
J. Ravi
Assistant Professor, Department of
Mathematics, Vivekanandha
College for Women,
Tiruchengode, Tamilnadu, India
Definition 2.2
Let X be a linear space and let f  X . The norm of f
is
denoted
f 
by
f
and
it
is
defined
as,
f, f.
Definition 2.3
Let X be a linear space. A real-valued
real
functional‖ . ‖on
X is called a norm provided for each f and g in X and
each real number  ,
f  0 and f  0 if and only if f  0 ,
i)
f g  f  g ,
ii)
iii)  f   f .By a normed linear space we mean
a linear space together with a norm.
Definition 2.4
Let f : X  R be a measurable function on a measure
space  X , A,   . The essential supremem of f on X is
ess sup f
=
inf{𝑎 ∈ ℝ:
ℝ 𝜇[𝑥𝜖𝑋: 𝑓(𝑥) > 𝑎] = 0}.
X


Definition 2.1
Let X be a measure space. The measurable function fis
said to be in Lp if it is p-integrable,
integrable, that is, if
Equivalently, ess sup f  inf sup g : g  f po int wisea.e.

Thus, the essential supremum of a function depends
only on its  -a.e.
a.e. equivalence class.
f
f
p
Lp
d    . The Lp


f
p
d

1/ p
norm of f is defined by
X
X
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Definition 2.4
Let  X , A,   be a measure space. The space L  X 
consists of point wise a.e.- equivalence classes of
essentially bounded measurable functions f : X  R
with norm f
L
 ess sup f . In future, we will write
X
ess sup f =sup f .
Definition 2.5
If p and q are positive real numbers such that
1 1
p  q  pq
or
equivalently,   1
,
p q
1  p  ,1  q    .Then we call p and q a pair of
Young’s inequality3.1
a p bq
 , with equality
p q
occurring if and only if a p 1  b .
for any a, b  0 , we have ab 
Holder inequality 3.2
Let 1  p   , and let
in the mean of order P if each f n belongs to LP and
f  fn
p
0.
1 1
  1 .If
p q
then fg  L1  E  and
 fg  
E
Lemma 3.3
Let 1  p   .
f g
p
p
 2p
fg  f
If f , g  L p ,
f
p
p
p
g q.
E
 g
p
p
then
f  g  Lp
. consequently,
LP
and
is
a
vector space.
Note:
Convergence in L is nearly uniform convergence.
Definition 2.7
A linear functional on a linear space X is a real-valued
function T on X such that for g and h in X and 𝛼 and
𝛽 real numbers,
T  .g  .h   .T  g   .T  h  .
Note: 2.8
1. The collection of linear functionals on a linear
space is itself a linear space.
2. A linear functional is bounded if and only if it is
continuous.
Definition 2.9
Let X and Y be a normed linear spaces. An isometric
isomorphism of X into Y is a one-one linear
transformation f : X  Y such that,
f  x  x ,
q be defined by
f  Lp  E  and g  Lq  E 
conjugate exponents.
Definition 2.6
A sequence of function x f n is said to converge to f
1 1
  1 .Then,
p q
Let 1  p   and let q be defined by
for all x  X .
3. WEAK SEQUENTIAL CONVERGENCE ON
Lp SPACE
Some basic inequalities and results have been
discussed.
Minkowski’s Inequality 3.4
Let 1  p   and let f , g  L p .Then, f  g  L p and
f g
p
 f
p
 g
p
.
Definition 3.5
Let X be a normed linear space. Then the collection of
bounded linear functionals on X is a linear space on
which ‖ . ‖∗ is a norm.This normed linear space is
called the dual space of X and denoted by X * .
Definition 3.6
Let X be a normed linear space. A sequence
 fn 
in X
is said to converges weakly in X to f in X provided
lim T
n 
*
 f n   T  f  for all T  X .We write  f n  ⟶ f
in X to mean that f and each f n belong to X and
 fn  converges weakly in X to
f.
Definition 3.7
A subset K of a normed linear space X is said to be
weakly sequentially compact in X provided every
sequence  f n  in K has a subsequence that converges
weakly to f  K .
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Theorem 3.8
Let E be a measurable set and 1  p   .
 fn 
and f
f in Lp  E  . Then  f n  is bounded in Lp  E 
p
 lim inf f n
p
Since  k .  f k 
*
Suppose
.
q

1
for all k , the sequence of partial
3k

sums of the series
  . f 
k 1
Lq  E  .
*
k
k
is a Cauchy sequencein
Proof:
Let q be a conjugate of p and f * the conjugate function
of f . We first establish the right-hand side on the above
inequality. We infer from Holder’s Inequality that
We know that, ‟If X is a measure space and 1  p   ,
f
Define the function g  L  E  by g 
*
. fn  f
E
f .
*
. fn p for all n .
q
q
E
Since
 fn 
converges weakly to f and f * belongs to
Lq  E  ,
n
f
p
 lim inf fn
E
p
By contradiction to show that
 fn 
is bounded in
 f  is unbounded. Without loss of
L  E  . Assume
p
n p
generality, by possibly taking scalar multiples of a
subsequence, we suppose
1
f n p  n.3n for all n ⇒ n f n p  n forall n .⟶①
3
We inductively select a sequence of real numbers  k 
1
1
for each k . Define 1  . If n is a
k
3
3
natural number for which 1 ,  2 ,....n have been
defined. Define
for which  k  

*
1
1
 n 1  n 1 if  k  fn   . fn1  0 and  n 1   n 1 if
3
3

E  k 1
the above integral is negative.
Therefore, by ① and the definition of conjugate
function,
n

1
n
*
k  fk  . fn  n fn
E 
3
k 1

p
n
*
 k . fk   . fn  n ⟶②
E 
k 1

and n . f n 
*
q

  . f 
k 1
*
k
k
.
We infer from the triangle inequality, ②, and Holder’s
Inequality that

*
k . fk   . fn
E g. fn  E 
k 1

f p   f *. fn  lim  f *. fn
E

Fix a natural number n .
fn  fn p for all n .
*
then Lp  X  is complete” [2] tells us that Lq  E  is
complete.
 n
   

*
*
   k . fk   . fn       k . fk   . fn 
   E kn1
 
 E  k1
n

*
*
  k . fk  . fn     k . fk  . fn


E  k1
E kn1
 
*
 n     k . fk   . fn

E k n1
  1
1 1
 n    k  . fn p  n  n . . f n
3 2
k n1 3 
 g. f
E
n

p
n
.
2
This is a contradiction because, the sequence  f n 
converges weakly in Lp  E  and g belongs to Lq  E  .


The sequence of real numbers   g . f n  converges and
E

therefore is bounded.
Hence  f n  is bounded in Lp .
Corollary 3.9
Let E be a measurable set, 1  p   and q the conjugate
of p . Suppose  f n  converges weakly to f in Lp  E 
1
for all n .
3n
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
 gn 
and
converges strongly to g in Lq  E  .Then
lim  gn . fn   g. f
n
E
E
E
 g .f   g. f   g .f   g.f   g.f   g.f
n
n
n
E
n
n
E
n
E
E
   gn  g . fn   g. fn   g. f
E
E
E
n
n
E
n
E
n
E
E
n
E
n
n
E
E
E
n
n 
E
E
n



E
E
.
E
E
E
E
E
E

E
 f
n
E
 f  .  g 0  g     f n  f  .g
E
 f .g   f .g
n
0
E
0
E
Since
 fn 
 fn  f
p
. g0  g q   f n .g   f .g .
E
E
is bounded in L  E  and the linear span of
p
is dense in Lq  E  , there is a function g in this linear
span for which
. g0  g
p


q
lim  fn .g   f .g .
n
E
E
Therefore, there is a natural number N for which
E


 f .g   f .g  2 if n  N .It is clear that②holds for
gn . fn  g. f .
It follows that lim
n
E
E
for all n .
2
We infer from ①, the linearity of integration, and the
linearity of convergence for sequences of real numbers,
that
E

0
E
fn  f
lim  gn . fn   g. f  0
n
n
E
and therefore, by using Holder’s inequality,
E
E
E
E
0
E
lim gn. fn   g. f  0
n
E
 f .g   f .g
lim  gn. fn   g. f  C.0   g. f   g. f
E
n
E
E
E
Substitute above inequalities in equation ① , we get
n
0
E
E
E
E
lim gn  g q  0and lim  g. fn   g. f
0
 fn .g0   f .g0    fn  f  . g0  g     fn  f  .g
From these inequalities and the fact that both
n
E
E
n
E
n
E
E
lim  gn. fn   g. f  C.lim gn  g q  lim  g. fn   g. f ⟶①
E
0
E
   f n  f .g0    fn  f  .g    fn  f  .g
Taking limit on both sides, we get

lim  gn . fn   g. f  lim C. gn  g q   g. fn   g. f
n
n

E
E
E
E
n
0
  f n .g 0   f . g 0   f n . g   f .g   f n . g   f . g
 C. gn  g q   g. fn   g.f for all n
E
  if n  N . ⟶②
0
E
n
n
E
 g .f   g. f
E
n
0
E
 g . f   g. f   g  g.f   g. f   g. f
n
n
E
 f .g   f .g   f .g   f .g   f .g   f .g   f .g   f .g
E
Therefore, by Holder’s Inequality,
n
 f .g   f . g
Observe that for any g  Lq  E  and natural number n ,
 g . f   g. f   g  g.f   g. f   g. f
n
E
must find a natural number N for which
E
E
E
lim  fn .g0   f .g0 .Let   0 . We
n
We show that
E
According to the preceding theorem, there is a constant
C  0 for which f n p  C for all n .
Consider
 f .g
for all g  𝓕.⟶①
Proof:
Assume that ① holds. To verify weak convergence, let
g 0 belong to Lq  E  .
Proof:
For each index n
E

f n .g 
f in Lp  E  if and only if lim
n
 fn 
n
E
Proposition 3.10
Let E be a measurable set, 1  p   , and q
E
E
this choice of N .Therefore lim  f n . g   f . g is true
n 
the
conjugate of p .Assume𝓕 is a subset of Lq  E  whose
E
E
for all g  𝓕.
linear span is dense in Lq  E  .Let  f n  be a bounded
sequence in Lp  E 
and f belong to Lp  E  . Then
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
4.
DUALITY ON Lp SPACE
Theorem 4.1
Suppose that
 X , A,   is
If f  Lp  X  ,
a measure space and
then F  g    fg
1
1 p   .
d ,
defines a bounded linear functional F : Lp  X   R ,
and F
 f
Lp*
1
Lp
B
. Then g  L1  X  with g
  B
Let g   sgn f 
.If X is a   finite, then the same
result holds for p  1 .
F g  f
1
Lp
g
Lp
which implies that F is a bounded
linear functional on L with F Lp*  f L .
In proving the reverse inequality, we may assume that
f  0 (otherwise the result is trivial).
First, suppose that 1  p   .
P
p1
 f
Let g   sgn f  
 f Lp1
and



p1 / p
1
.Then g  Lp , since f  Lp ,
g Lp 1.
Also,
 f
F  g     sgn f  f 
 f p1

L
Since
p1
 p1  1 ,
p
since



1
p 1
d  f
g Lp 1, we have
F Lp*  f
F
p*
L
1
Lp
.
 F  g  ,so that
F g  
1
f d  f
  B  B
It follows that
F L1*  f
1
Lp
.
F Lp* is actually attained for a suitable
function g .
Suppose that p  1 and X is   finite. For   0 , let

L
L
.
, therefore F L1*  f
L
since   0 is arbitrary.
This theorem shows that the map F  J  f  defined
by J : Lp  X   Lp  X  ,
*
J  f  : g   fg
1
d  , Is a isometry from Lp into Lp* .
Note: 4.2
The following result is that J is onto when 1  p   ,
meaning that every bounded linear functional on LP
1
arises in this way from an Lp - function. The proof is
based on that if F : Lp  X   R is a bounded linear
functional on Lp  X  , then   E   F   E  defines an
absolutely continuous measure on  X , A,   .
Lemma4.3
Let  X , A,  
be a   finite measure space and
1  p   . For f an integrable function over X,
suppose there is an M  0 such that for every simple
function g on X that vanishes outside of a set of finite
measure,
 fg
d   M . g p . …(1). Then f belongs
X
to L  X ,   , where q is conjugate of p . Moreover,
q
If p   , we get the same conclusion by taking
g  sgn f  L . Thus, in these cases the supremum
defining
1,
and
1
Proof:
From Holder’s inequality, we have for 1  p   that
L1
A  x  X : f  x  f
L

 .
Then 0    A   .Moreover, since X is   finite,
there is an increasing sequence of sets An of finite
measure whose union is A such that   An     A . So
we can find a subset B  A such that 0    B    .
f
q
M .
Proof:
First consider the case p  1 .Since f is a non-negative
measurable function and the measure space is   finite,
according to the Simple Approximation Theorem, there
is a sequence of simple functions n  , each of which
vanishes outside of a set of finite measure, that
converges point wise on X to f and   n  f on E
for all n .
q
Since  nq  converges point wise on X to f .
We know that,” If  X , A,   is a measure space and
 fn 
a sequence of non-negative measurable functions
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
on X for which
 fn 
f point wise a.e. on X and if f
f
is measurable. Then
d   lim inf
X
 f d  .” [3]
Theorem 5.1
Let X be a separable normed linear space and Tn  a
q
sequence in its dual space X * that is bounded, that is,
n
X
The above statement tells us that to show that f is
integrable and
f
M
q
it suffices to show that
X
Fix a natural number n. To verify (1), we estimate the
functional values of  nq as follows:
 nq   n  nq 1  f .nq 1

q
n
q 1
n
X
d   M g n p . ….....................................(4)
Since p and
q
are
therefore  g n d     n 
p q 1
p
X
p  q  1  q
conjugate,
X
and
d     nq d 
X
1/ p


Thus we may rewrite (4) as   d   M .    nq d  
X
X

q
n
For each ,  n is a simple function that vanishes
outside of a set of finite measure and therefore it is
integrable. Thus the preceding integral inequality may
be rewritten as
q
n
11/ p
 q 
  n d  
X

T  of
Tn  and
X*
for
 f   T  f  ….(2) for all f
in X .
nk
lim Tnk
k 
T in
which
Let
f 
j j1 be
a countable subset of X that is dense in
X .We infer from (1) that the sequence of real numbers
Tn  f1  is bounded. Therefore, by the Bolzano-
WeierstrassTheorem,“Every bounded sequence of real
numbers has a convergent subsequence.” there is a
strictly increasing sequence of integers s 1, n  and a
d    f .g n d 
X
f ……(1)

Define the simple function g n by g n  sgn  f  .
on X.
We infer from (1) and (2) that
X
  M.
Proof:
 nq  f .sgn  f  .nq 1 on X…............................(3)
q
n
there is an M  0 for which Tn  f
for all f in X and all n .Then there is subsequence
q
q
 n d   M for all n…................................(2)

5. WEAK SEQUENTIAL COMPACTNESS
number a1 for which lim Ts1,n   f1   a1 .
n 
We again use (1) to conclude that the sequence of real


numbers Ts 1, n   f 2  is bounded and so again by the
Bolzano-Weierstrass Theorem, there is a subsequence
s  2, n  of s 1, n  and a number for a2 which
lim Ts 2,n  f2   a2 .We inductively continue this
n
selection process to obtain a countable collection of
strictly increasing sequences of natural numbers


s j, n

j1
and a sequence of real numbers a j 
such that for each j , s  j 1, n  is a subsequence of
M.
1 1
 , we have verified (2).
p q
It remains to consider the case p  1 . We must show
that M is an essential upper bound for f .
We argue by contradiction.
If M is not an essential upper bound, then there is some
Since 1 


∈>0 for which the set X   x  X : f  x   M   has
nonzero measure.
Since X is   finite, we may choose a subset of X 
with finite positive measure.
If we let g be the characteristic function of such a set
we contradict (1).
T  f j   aj .
s  j, n  , and lim
n s j,n 
For each index k , define nk  s  k , k  . Then for each j ,
nk k  j

is a subsequence of
s  j, k 
lim Tnk  f j   a j for all j .Since T n
k 
k
 is
and hence
bounded in
  f  is a Cauchy sequence for each
a dense subset of X , T  f  is Cauchy for all
X * and T n k
nk
f is
f in
X . The real numbers are complete. Therefore we may
define
T  f   limTnk  f  for all f  X .
k 
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Page: 1147
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Since each Tnk is linear, the limit functional T is
linear.
  M . f for all k and
T  f   lim Tn  f   M . f for all f  X
k 
Since
Tnk  f
all f  X ,
k
Therefore T is bounded.
Theorem 5.3
Let E be a measurable set and 1  p   . Then every
bounded sequence in Lp  E  has a subsequence that
converges weakly in Lp  E  to a function in Lp  E  .
Proof:
Let q be the conjugate of p .
Let  f n  be a bounded
sequence in Lp  E  . Define X  Lq  E  .
Let n be a natural number. Define the functional Tn on
Tn  g    fn .g for g in X  L  E  .
q
E
We use the statement, ‟ Let E be a measurable set,
1  p   , q the conjugate of p and g belong to
Lq  E  . Define the functional T on Lp  E  by
T  f    g. f for all f  L  E  .
p
E
Then T is a bounded linear functional on Lp  E  and
T
*
 g q .”
Now p and q interchanged and the observation that p
is the conjugate of q , tells us that each Tn is a bounded
linear functional on X and
Since
 fn 
Tn *  fn
p
is a bounded sequence in Lp  E  , Tn  is a
for which
Tn  f   M . f for all f in X and all n ”.
 
Then there is subsequence Tnk of Tn  and T in X *
 f   T  f  for all f in
for which lim Tnk
k 
is
Note 5.2:
The following theorem tells the existence of week
sequential compactness on Lp space
X by
bounded, that is, there is an M  0
a
 
subsequence Tnk
and T  X *
X .”There
such
that
limTnk  g   T  g  for all g in X  Lq  E  …(1)
k 
The Riesz Representation Theorem, with p and q
interchanged, tells us that there is a function f in
Lp  E  for which T  g  
 f .g for all g in X  L  E  .
q
E
But
(1)means
that lim
k 

f
nk
E
.g   f .g for all g in
E

g. f n  g. f
X  Lq  E  lim
. n E
E
According to‟ Let E be a measurable set, 1  p  
and q the conjugate of p . Then  f n   f in Lp  E  if
and only iffor all g  Lq  E  .”
  converges weakly to f
Therefore f nk
in Lp  E  .
Hence every bounded sequence in Lp  E  has a
subsequence that converges weakly in Lp  E  to a
function in Lp  E  .
6. CONCLUSION
In this paper the duality, weak convergences and weak
sequential compactness on 𝐿 space has been discussed.
A new generalized 𝐿
space viz. theorem of
compactness on 𝐿 spaceare investigated. We hope that
the space will be a powerful tool to study the various
properties of lebesgue space and this endeavor is a
small step towards that goal.
bounded sequence in X * .
Moreover, according to ‟ A function f on a closed,
REFERENCES
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 a, b 
is absolutely continuous on
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 a, b .” [3]
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Therefore, by ‟Let X be a separable normed linear
space and Tn  a sequence in its dual space X * that is
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Macmillan, New York, 1988.
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017
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Functional
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017
Page: 1149