Analyzation of Weak Convergence on L^p Space

1. International Research Research and Development (IJTSRD) International Open Access Journal International Open Access Journal International Journal of Trend in Scientific Scientific (IJTSRD) ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume ISSN No: 2456 | www.ijtsrd.com | Volume - 2 | Issue – 1 Analyzation of Weak Convergence on L^p Space Analyzation of Weak Convergence on L^p Space Analyzation of Weak Convergence on L^p Space R. Ranjitha S. Dickson J. Ravi Research Scholar, Department of Mathematics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Tamilnadu, India Assistant Professor, Department of Assistant Professor, Department of Mathematics, Vivekanandha College for Women, Tiruchengode, Tamilnadu, India Assistant Professor, Department of Mathematics, Vivekanandha College for Women, Assistant Professor, Department of Mathematics, Vivekanandha College for Women, Tiruchengode, Tamilnadu, India Tiruchengode, Tamilnadu, India Tiruchengode, Tami ABSTRACT In this paper a study on duality, weak convergences and weak sequential compactness on ?? space. A new generalized ?? space is investigated from the weak sequential convergence. Some more special types and properties of ?? are presented. Few relevant examples are also included to justify the proposed notions. Keywords: Dual space, Linear functional, Weak Convergence, Compact paper a study on duality, weak convergences and Definition 2.2 Let X be a linear space and let is denoted by  , f f f  Definition 2.3 Let X be a linear space. A real X is called a norm provided for each each real number  , i) 0 f  and f  if and only if       .By a normed linear space 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 f ess = inf{? ∈ ℝ  X Thus, the essential supremum of a function depends only on its  -a.e. equivalence class.  space. A new Let X be a linear space and let f . The norm of f X is investigated from the weak Some more special types and are presented. Few relevant examples are also included to justify the proposed notions. f and and it it is is defined defined as, as,  . Dual space, Linear functional, Weak linear space. A real-valued functional‖ .‖on X is called a norm provided for each f and gin X and 1.INTRODUCTION f  , 0 if and only if 0 p The part of a program to formulate for functional and mappings defined on infinite dimensional spaces appropriate versions of properties possessed by functionals and mappings defined on finite dimensional spaces. In this paper, we study the weak convergence and weak compactness on the so-called Lebe spaces (or L – spaces as they are usually called). 2.PRELIMINARIES L spaces were introduced by FrigyesRiesz 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 ings defined on finite dimensional In this paper, we study the weak convergence f g f g ii) , f f iii) normed linear space we mean a linear space together with a norm. called Lebesgue be a measurable function on a measure essential supremem of f on X is ℝ:?[???:?(?) > ?] = 0}. p spaces as they are usually called). Xsup  Definition 2.1 Let X be a measure space. The measurable function said to be in L if it is p-integrable, d f      L Equivalently,     ess ess sup sup X f f inf sup inf sup g g : : g g f po f po int int wisea e wisea e . . . . measure space. The measurable function fis integrable, that is, if X X p Thus, the essential supremum of a function depends a.e. equivalence class. p p . The L norm of f is defined by is defined by  1/ p  p d f f p @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Dec 2017 Page: 1142

2. Definition 2.4 Let  X A  be a measure space. The space consists of point wise a.e.- equivalence classes of essentially bounded measurable functions sup L X ess sup f =sup f . Definition 2.5 If p and q are positive real numbers such that equivalently,1 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Young’s inequality3.1     X   and let qbe defined by 1 1 q , , L   .Then, Let1 p 1 p X  R f : p q a b q   a b  ab for any , , we have , with equality 0   p f ess f with norm . In future, we will write  p 1 occurring if and only if Holder inequality 3.2   , and let qbe defined by 1 . a b 1 q   .If  Let 1 1 p p 1 q   E   E   f L g L and p q     pq or 1 , p q p    conjugate exponents. Definition 2.6 A sequence of function             .Then we call p and q a pair of fg L E fg fg f g 1 p ,1 q then and . 1 p q E E Lemma 3.3 Let1       . If ,  , then and f g L  p f g L p nf x is said to converge to f p p p p p    p P f g 2 f g . consequently, L is a P in the mean of order P if each 0 n p f f 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 ? and ? real numbers,     T . . .T . g h 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  , for all x X  3.WEAK SEQUENTIAL CONVERGENCE ON p L SPACE Some basic inequalities and results have been discussed. nf belongs to L and p p   . vector space. Minkowski’s Inequality 3.4 Let 1 p    and let ,      .Then, and f g L f g L p p  . f g f g p p 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 Definition 3.6 Let X be a normed linear space. A sequence   is said to converges weakly in X to f in X provided     lim n n   in X to mean that f and each   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   in K has a subsequence that converges weakly to f K  .   . * X . nf in X .We write   * nf for all T X ⟶ f  T f T f nf belong to X and nf . nf @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1143

3. Theorem 3.8 LetE be a measurable set and 1   f  in   L and lim inf p f Proof: Letqbe a conjugate of p and of f . We first establish the right-hand side on the above inequality. We infer from Holder’s Inequality that * * . . n n p q E *. n n p E Since   converges weakly to f and   f f f  n E lim inf n p p f f By contradiction to show that     L n p f is unbounded. Without loss of generality, by possibly taking scalar multiples of a subsequence, we suppose for alln⇒1 3 We inductively select a sequence of real numbers   1 3 natural number for which defined. Define   1 k E    the above integral is negative. Therefore, by ① ① and the definition of conjugate function, 1 . 3 k E             . 3 q International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 1 3     *   for allk , the sequence of partial Since . f   . Suppose is bounded in  p k k k q E . Then  f   E p p nf nf L   *  . f sums of the series is a Cauchy sequencein k k  .  k 1 n p   qL E . We know that, ‟If X is a measure space and1 then   LpX is complete” [2] tells us that complete.    , p   * f the conjugate function qL E is       *      g . f f f f f  q g L E for all n. Define the function by . k k  k 1 Fix a natural number n. We infer from the triangle inequality, ② Inequality that                                         1 . 3 k n     n  .  f f f for all n. ②, and Holder’s * nf f belongs to qL E ,       f   * g f . . . f n k k n  *.  * lim  f . f  k 1 E E n n p E          n     f   f   * *    . . f . . f  k k n k k n k n     k 1 1 E E    nf is bounded in     n     . f   . f E . Assume     * *  . f . f p k k n k k n  k n   k 1 1 E E     f  * n . . f k k n  foralln.⟶①  k n    .3n n ⟶① 1 f n f E  n n n p p    1 1 . . 3 2     n f k n f n n k n p p 1 1 3      . If nis a for eachk . Define for which k 1 k  g f . n    have been 2 , ,....n E 1 2 This is a contradiction because, the sequence  converges weakly in   L E and g belongs to nf     n   * 1 n 1 n   f . f 0      if and if   p qL E . k n n 1   n 1 n 1   1 1 3 3        g f . The sequence of real numbers converges and n E therefore is bounded. Hence  Corollary 3.9 Let E be a measurable set,1 of p . Suppose      n    f p nf  is bounded in L . *   f f k k n n n p 1 n     andq the conjugate converges weakly to f in   * p  . f . f n ⟶② ⟶② k k n   E p nf L  k 1 E 1 *   f for all n. and n n n @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1144

4. and   lim n E Proof: For each index n . n n E  n E According to the preceding theorem, there is a constant 0 C  for which n p f C for all n. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470     qL E .Then  n g g f converges strongly to g in       E if and only if lim f g . f g .  p nf f L in n  n  . g f . E E n n for all g ?.⟶① Proof: Assume that ① 0 g belong to ⟶①  E ① holds. To verify weak convergence, let   lim . . n n E E must find a natural number N for which     g L E                   0 . . n n E E E    0 0 0 . . . n n E E E    qL E .            g f g f . g f . g f . g f . g f . n n n n    f g f g   . We 0 We show that .Let E  g .  E  E E E 0 0       g f g f . g f . n n E E    f g . f g .  if n N . ⟶② ⟶②  n 0 0 E E q and natural number n,     . f g  Observe that for any             g f . g f . g g f . g f . g f . Consider n n n n n      f g . f g . f g . f g . f g . f g . f g . f g . n 0 0 n 0 0 n n E E E E E E  E  E E E E E E Therefore, by Holder’s Inequality,        f g . f g . f g . f g . f g . f g . n 0 0 n n   E  E f g E E f g E  E        g f . g f . g g f . g f . g f .        f f f n n n n n n E E E E E              f g f g f f g g f f g .     for alln      g f . g f . C g . g g f . g f . n n n n n q E E E E E    .           Taking limit on both sides, we get f g . f g . f f g g f f . g n 0 0 n 0 n      E E E E     and therefore, by using Holder’s inequality,   Since   is bounded in is dense in   span for which          lim n  g f . g f . lim n  C g    . g g f . g f . n n n n q E E E E   .       f g . f g . f f . g g f g . f g . n 0 0 n 0 n p q          ⟶① ⟶① lim n  g f . g f . C .lim n  g g lim n  g f . g f . E E E E n n n n q   E and the linear span of p nf L E E E E From these inequalities and the fact that both lim g 0 n q n  n E Substitute above inequalities in equation ①    qL E , there is a function g in this linear     andlim  g g f . g f . . n  E ① , we get f f . g g for alln. n 0 p q 2      We infer from ① linearity of convergence for sequences of real numbers, that lim . . n n E E Therefore, there is a natural number N for which      if n N ①, the linearity of integration, and the lim n  g f . g f . C .0 g f . g f . n n E E E E     lim n  g f . g f . 0    f g f g n n . E E      lim n  g f . g f . 0 n n E E    It follows thatlim g f . g f . f g . f g .  . .It is clear that② ②holds for n n n 2  n E E E E Proposition 3.10 LetE be a measurable set,1 conjugate of p .Assume? is a subset of    this choice of N .Thereforelim f g . f g . is true n  n E E   , and q the   be a bounded   L E . Then p for all g ?. qL E whose qL E .Let    nf linear span is dense in   E and f belong to p p L sequence in @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1145

5. 4. DUALITY ON Theorem 4.1 Suppose that     . International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470  p L SPACE       ,  1 g L X g 1 B g sgn f . Let Then with   B 1  L and   F  X A  is a measure space and   L X  , , , 1      . g f d f then   F g   B  p 1 d  p  If , f fg 1 p L B   X R ,  , therefore  F L : defines a bounded linear functional F f  result holds for 1 p  . Proof: From Holder’s inequality, we have for 1   p p L L F g f g F f F f It follows that since   is arbitrary. This theorem shows that the map   : J L X L    J : f g fg Note: 4.2 The following result is thatJ is onto when 1 meaning that every bounded linear functional on arises in this way from an based on that if   : F L X R is a bounded linear functional on   absolutely continuous measure on  Lemma4.3 Let  , , X A  be a finite measure space and 1 p    . For f an integrable function over X, suppose there is an 0 M  such that for every simple function g on X that vanishes outside of a set of finite   1 * 1 * L L L L .If X is a finite, then the same 0 and 1 p * p L L   J f  F defined   X * 1 p p by ,  1 d p p * L into Is a isometry from L . ,    that p  which implies that F is a bounded  1 F f    , L p P linear functional on In proving the reverse inequality, we may assume that 0 f  (otherwise the result is trivial). First, suppose that 1 p    . L with . 1 p * p L L P 1 p L - function. The proof is p LpX , then     1 p p /   X A  .  E F defines an         f E    1 g sgn f   p p f L Let .Then g L , since ,  , , f 1 p L  . g 1 and p L 1 p p   , 1 p 1 Also, since 1  p 1         f       f 1. p L   F g sgn f f d f 1 p L    fg d M g . . measure, …(1). Then f belongs    , we have  g 1 F F g p Since ,so that p p * X L L   qX   L , , to where q is conjugate of p . Moreover,  F f . 1 * p L p L . f M If p   , we get the same conclusion by taking sgn g f L   . Thus, in these cases the supremum F is actually attained for a suitable function g . Suppose that p 1  and X is finite. For    : L A x X f x f Then   0 A     .Moreover, since X is finite, there is an increasing sequence of sets measure whose union is A such that  we can find a subset B A  q Proof: First consider the case measurable function and the measure space is finite, according to the Simple Approximation Theorem, there is a sequence of simple functions , vanishes outside of a set of finite measure, that converges point wise on X to f and for all n. Since   n  converges point wise on X to We know that,” If , , X A  is a measure space and   a sequence of non-negative measurable functions defining p  .Since f is a non-negative 1 p * L   , let 0  each of which  n     .      f on E n n A of finite   A     B    . q q . f   n A 0 . So   such that nf @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1146

6. on X for which   International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 5.WEAK SEQUENTIAL COMPACTNESS  nf f point wise a.e. on X and if f Theorem 5.1 Let X be a separable normed linear space and  sequence in its dual space there is an 0 M  for which for all f in X and all n.Then there is subsequence   of T in     lim k n k  Proof: Let   1 j X .We infer from (1) that the sequence of real numbers    1 f 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      is measurable. Then f d liminf f d .” [3] n n T a X X q * X that is bounded, that is,   f M  The above statement tells us that to show that  is f n T . f ……(1) integrable and it suffices to show that f M q     q n q d M for all n…................................(2)    k n T * n T and X for which X Fix a natural number n. To verify (1), we estimate the functional values of n  as follows:     . n f      .sgn . n f f   on X…............................(3) Define the simple function on X. We infer from (1) and (2) that  T f T f ….(2) for all f in X . q   q 1 q n q n 1 n  q 1 q n  f be a countable subset of X that is dense in   f    j q n 1 g sgn . n g by n  n T     q n d f . g d n   X  X   s 1, n and a    q n d M g . ….....................................(4)   f  lim n  T a n p number 1a for which .  1 1 s 1, n X   1   and p q q Since p and q are conjugate, We again use (1) to conclude that the sequence of real  2 1, s n T f is bounded and so again by the Bolzano-Weierstrass Theorem, there is a subsequence    2, s n of   1, s n and a number for   2 2 2, lim s n n  selection process to obtain a countable collection of strictly increasing sequences of natural numbers  1 j   j 1, s     j, s n , and  j, lim j s n n  For each index k , define   k k j n   lim j j k      k n T f is a Cauchy sequence for each f is    k n T X . The real numbers are complete. Therefore we may define     lim kn k     numbers   1     p p q n        q n  therefore g d d d n X X X    1/ p       2a which        q n q n d M . d Thus we may rewrite (4) as  T f a .We inductively continue this X X   is a simple function that vanishes q n For each n, outside of a set of finite measure and therefore it is integrable. Thus the preceding integral inequality may be rewritten as 1 1/ p M d       1 1 1 p q It remains to consider the case 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 ∈>0 for which the set  X 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).       s j, n  and a sequence of real numbers    n   T f a  s k k  ja     q n . such that for each j ,  is a subsequence of X  . j   Since , we have verified (2).  kn ,  . Then for each j ,  j,k and hence  is bounded in p  . We must show 1    s  is a subsequence of  for all j .Since T f a T k n n k    f x     has x X : M * X and  a dense subset of X , f is Cauchy for all f in  T f T f  for all f . X @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1147

7. Since each linear. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 M  for which 0 bounded, that is, there is an   . f M f  Then there is subsequence   lim k n k  subsequence      lim kn k  The Riesz Representation Theorem, with p and q interchanged, tells us that there is a function f in E for which   E But (1)means thatlim k E   X L E  . n n E E According to‟ Let E be a measurable set, 1 and qthe conjugate of p . Then  and only iffor all   g L E  .” Therefore converges weakly to f in Hence every bounded sequence in k n T is linear, the limit functional T is n T for all f in X and all n”.   f    f k n T  M . f of    for allk and all f Since   T f , X k n T   f * n T and T in X   f  lim k  T M f .   T T for all f X for which for all f in X .”There kn ThereforeT is bounded. Note 5.2: The following theorem tells the existence of week sequential compactness on Lp space Theorem 5.3 Let E be a measurable set and 1 bounded sequence in   L E has a subsequence that converges weakly in   L E to a function in Proof: Let qbe the conjugate of p . Let  sequence in   L E . Define Let n be a natural number. Define the functional   . n n E We use the statement, ‟ Let E be a measurable set, 1 p    , qthe conjugate of p and g belong to     g. E ThenT is a bounded linear functional on  .” knT  * is a and T X such that    T g T g  q X L E for all g in …(1)  T g f g .        . Then every p  p q L X L E for all g in . p     f g . f g .   E . p p L for all g in kn  E   .lim  g f g f . q nf be a bounded  L   p   E if    p q X L E .  p nf f in nT on q  T g f g    q X L E  X by for g in .   E .   E has a   L E to a k nf p L p L p subsequence that converges weakly in   L E . 6.CONCLUSION In this paper the duality, weak convergences and weak sequential compactness on ?? space has been discussed. A new generalized ?? space viz. theorem of compactness on ?? 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. REFERENCES   E by qL E . Define the functional T on  p L p function in   E  p f L T f f for all .   E and p L T g * q Now p andq interchanged and the observation that p is the conjugate of q, tells us that each nT is a bounded  T f linear functional on X and Since   is a bounded sequence in bounded sequence in Moreover, according to ‟ A function f on a closed, bounded interval   , a b is absolutely continuous on   , a b if and if only if it is an indefinite integral over   , a b .” [3] Since1 q   ,   X L E  is separable. Therefore, by ‟Let X be a separable normed linear space and  n n p L *   E ,   p nf n T is a * X . 1.Novinger, W.P [1992], Mean convergence in Lp spaces, Proceedings of the American Mathematical Society, 34, 627-628. 2.Walter Rudin [1966], Real and complex analysis, 3rdedition, Tata McGraw Hill, New York, 1987. 3.Roydon,H.L[1963], Real Analysis, 3rd edition, Macmillan, New York, 1988. q * n T a sequence in its dual space X that is @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1148

8. 4.Jain P.K, Gupta V.P and Pankaj Jain, Lebesgue Measure And Integration, 2nd edition, New Age International (P) Limited, New Delhi. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 5.Carothers University press, UK. N.L, Real Analysis, Cambridge 6.Karunakaran .V, Real Analysis, Pearson, India, www.pearsoned.co.in/vkarunakaran. 7.Royden H.L, and Fitzpatrick P.M, Real Analysis, 4th edition, Pearson India Education Services Pvt. Ltd. India, www.peaeson.co.in. 8.FrigyesRiesz Analysis,Dover, 1990. and BelaSz-Nagy, Functional @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1149