TWO-NORM CONVERGENCE IN THE L1 SPACE IN THE SENSE OF VITALI
Ch. Rini Indrati
Department of Mathematics, Gadjah Mada University, Sekip Utara, Yogyakarta, Indonesia, rinii@ugm.ac.id or ch.rini indrati@yahoo.com
Abstract. In this paper we consider L1 as a two-norm space in the sense of Vitali and prove a representation for two-norm continuous functionals in the sense of Vitali defined on L1.
Key words and Phrases: two-norm, Vitali, representation, and L1.
Abstrak. Di dalam paper ini, L1 dipandang sebagai ruang two-norm dalam arti Vitali. Selanjutnya diberikan representasi untuk fungsional kontinu two-norm dalam arti Vitali.
Kata kunci: two-norm, Vitali, representasi, dan L1.
1. Introduction
Hildebrandt [2] and Khaing ([3], [5]) proved a representation theorem for BV [0, 1] by regarding BV [0, 1] as a two-norm space. They overcome the lack of a fact that the Banach dual of BV [0, 1] is not C[0, 1] if we endorse BV [0, 1] with its usual norm, namely, |f (0)| + V (f ; [0, 1]) where V (f ; [0, 1]) denotes the total variation of f on [0, 1]. Here, C[0, 1] and BV [0, 1] denote the space of all continuous functions on [0, 1]
and the space of all functions of bounded variation on [0, 1], respectively.
As we know, the Riesz representation theorem is well-known [9, 10]. If p and q are two real numbers, 1 < p, q < ∞ and 1p +1q = 1, the Banach dual of Lp[0, 1] is Lq[0, 1].
We also have that the Banach dual of L1[0, 1] which is L∞[0, 1]. However, the Banach dual of L∞[0, 1] is not L1[0, 1] if we endorse L∞[0, 1] with the usual norm, ess sup |f |, where ess sup |f | = inf{M : |f (x)| ≤ M a.e. on [0, 1]}.
In [4], Indrati proved a representation theorem for L∞[0, 1] by regrading L∞[0, 1]
as a two-norm space. Here we use a two-norm convergence of a sequence of functions {fn} as follows: a sequence of functions {fn} is said to be two-norm convergent in
2000 Mathematics Subject Classification: 90C27, 94A99 105
L∞[0, 1], if there exists a positive constant M such that kfnk ≤ M , for all n andR1
0 fng converges for all g ∈ L1. If we consider L∞as a two-norm space, we have the dual of L∞
is L1. The important thing in two-norm space in L∞ is the uniformly bounded criteria of the sequence. This idea brings the author to have a new kind of two-norm in L1by replacing the bounded criteria of the sequence by uniformly absolutely continuous which is introduced by Vitali [1]. This new criteria gives more general result in representation theorem on L1[0, 1].
In this paper, we consider L1, as a two-norm space in the sense of Vitali and prove a representation theorem for two-norm continuous linear functionals on L1in the sense of Vitali. The proof of the completeness will be done by Ascoli’s Theorem. Here some important facts that are used in the research.
A function G, defined on [0, 1], is said to be absolutely continuous if for every
² > 0 there is a δ > 0 such that q (D)X
|{G(v) − G(u)}| < ² whenever (D)P
|v − u| < δ, where D = {[u, v]} denotes a partial division of [0, 1] in which [u, v] stands for a typical interval in the partial division.
A sequence of functions {Gn} is said to be uniformly absolutely continuous on [0, 1], if for every ² > 0, there exists a constant δ > 0 such that
(D)X
|{Gn(v) − Gn(u)}| < ², for every n, whenever (D)P
|v − u| < δ, where D = {[u, v]} denotes a partial division of [0, 1] in which [u, v] stands for a typical interval in the partial division. We are using the notation of the Henstock integral [6, 8].
As we know that every monotone function on [0, 1] is differentiable almost ev-erywhere on [0, 1]. Since every bounded variation function can be represented as a difference of two increasing functions (Jordan’s decomposition) and every absolutely continuous function has bounded variation, we have Theorem 1.1.
Theorem 1.1. If F is absolutely continuous on [0, 1], then the function F0exists almost everywhere in [0, 1] and F0 is absolutely integrable on [0, 1].
Proof. See [6]. 2
In what follows, when we say absolutely integrable we mean Lebesgue inte-grable.
Theorem 1.2. If f is absolutely integrable on [0, 1], then the function F , defined on [0, 1] by
F (x) = Z x
0
f (t) dt, (1)
for every x ∈ [0, 1], is absolutely continuous. Moreover, F0= f a.e. in [0, 1].
We close this section by restate the Ascoli’s Theorem [11], however, we need to have some definitions which are needed in the theorem.
Definition 1.3. Let {Fn} be a sequence of functions defined on a set E. We say {Fn} is pointwise bounded on E if the sequence {Fn(x)} is bounded for every x ∈ E, that is if there is a finite-valued function φ defined on E such that
|Fn(x)| < φ(x), x ∈ E n = 1, 2, 3, . . . .
We say {Fn} is uniformly bounded on E if there is a positive number M such that
|Fn(x)| < M, x ∈ E n = 1, 2, 3, . . . .
Definition 1.4. Let {Fn} be a sequence of functions defined on a set E. We say {Fn} is equicontinuous on E if for every ² > 0, there is a η > 0, such that for every x, y ∈ E, if |x − y| < η, then
|Fn(x) − Fn(y)| < ², for every n.
It is clear that if a sequence of function {Fn} is uniformly absolutely continuous on [0, 1], then {Fn} is equicontinuous on [0, 1].
Here is the Ascoli’s Theorem in Theorem 1.5 below.
Theorem 1.5. If K is compact, {Fn} is a sequence of continuous functions on K, and {Fn} is pointwise bounded and equicontinuous on K, then
(i) {Fn} is uniformly bounded on K.
(ii) {Fn} contains a uniformly convergent subsequence.
2. Two-norm Convergence and Two-norm Continuous on L1 in the sense of Vitali
Let L1denote the space of all absolutely integrable functions on [0, 1].
A sequence {fn} of functions is said to be two-norm convergent in the sense of Vitali in L1, if there exists {Fn} ∈ U AC[0, 1] such that Fn0 = fn almost everywhere in [0, 1] for every n and
n→∞lim Z 1
0
fn(x)g(x) dx exists, for every g ∈ L∞.
By Theorem 1.2, for every n, if fn is absolutely integrable on [a, b], the function Fn defined on [0, 1] by
Fn(x) = Z x
0
fn(t) dt, (2)
for every x ∈ [0, 1], is absolutely continuous.
For every n, Fn is the primitive of an integrable function fn on [0, 1]. If there is an integrable function g on [0, 1] such that {fn} satisfies |fn(x)| ≤ g(x) for almost all x ∈ [0, 1] for all n, then {Fn} is uniformly absolutely continuous on [0, 1] [6]
We shall prove the completeness in Theorem 2.3. However, we need the big Sandwich Lemma. We state without proof the big Sandwich Lemma [7].
Lemma 2.1. If 0 ≤ an ≤ bkn for all n, k and
k→∞lim lim
n→∞bkn= 0 then limn→∞an= 0.
Lemma 2.2. If {Fn} is uniformly absolutely continuous on [0, 1] that converges uni-formly to a function F on [0, 1], then F is absolutely continuous on [a, b].
Proof. Let ² > 0 be given. By the hypothesis, there is an η > 0 such that for every collection D = {I} of non-overlapping intervals in [0, 1], if (D)P
|I| < η, we have (D)X
|Fn(I)| < ²
2, for all n.
Since {Fn} converges uniformly to F on [a, b], there is a positive integer number no
such that for all n ≥ nowe have
|Fn(x) − F (x)| < ² 2no+4.
Therefore, for every collection D = {I} = {Ii : i = 1, 2, . . . , m} of non-overlapping intervals in [0, 1], if (D)P
|I| < η, by taking p = max{no, m}, we have (D)X
|F (I)| ≤ (D)X
|F (I) − Fp(I)| + (D)X
|Fp(I)|
< (D)X ² 2p+4 + ²
2 < ² 2+ ²
2 = ². 2 In Theorem 2.3 and Lemma 2.6, we define
γt(x) =
½ 1 for 0 ≤ x < t 0 for t ≤ x ≤ 1.
Theorem 2.3. If {fn} is two-norm convergent in the sense of Vitali in L1, then there exists a function f ∈ L1, such that
Z 1
0
fng → Z 1
0
f g, as n → ∞, for every g ∈ L∞.
Proof. Since {fn} is two-norm convergent in the sense of Vitali, there exists a sequence {Fn} ∈ U AC such that Fn0 = fn almost everywhere in [0, 1] for every n and
n→∞lim Z 1
0
fng
exists. Since {Fn} is UAC on [0, 1], then {Fn} is equicontinuous on [0, 1] and uniformly bounded on [0, 1]. By Ascoli Theorem, there is a subsequence {Fnk} of {Fn} that is uniformly convergence, say to F on [0, 1]. We have F ∈ AC[0, 1]. therefore F0(x) exists for almost all x ∈ [0, 1] and F0 is absolutely integrable on [0, 1]. Let f be a function on [0, 1] such that f = F0 almost everywhere in [0, 1]. We have f ∈ L1. For every n, fn is
absolutely integrable on [0, 1], so fn is absolutely integrable on every closed subinterval of [0, 1]. That means fn is absolutely integrable on [0, x], for every x ∈ [0, 1]. Without lost generality, we may put Fn as the primitive function of fnfor every n. Furthermore, for every x ∈ [0, 1], Z x
0
fn(t)dt = Z 1
0
fn(t)γx(t)dt,
for every n. That means, {Fn(x)} is convergent for every x ∈ [0, 1]. Therefore, {Fn(x)}
is Cauchy sequence for every x ∈ [0, 1]. As corollary, {fn} is Cauchy sequence. Since {Fnk} converges uniformly to F on [0, 1], we have {fnk} converges to f almost every-where on [0, 1]. Therefore, {fn} converges to f almost everywhere on [0, 1]. Further-more, if g is a step function on [0, 1], we have
Z 1
0
fng → Z 1
0
f g, as n → ∞.
For every g ∈ L∞, fng, f g ∈ L1 and
| Z 1
0
fng − Z 1
0
f g| = | Z 1
0
(fn− f )g| ≤ Z 1
0
|fn− f | kgk∞ → 0 as n → ∞. 2 It is clear from the definition, if g is essentially bounded on [0, 1] and {fn} is two-norm convergent to f in L1 in the sense of Vitali, then
n→∞lim Z 1
0
fn(x)g(x)dx = Z 1
0
f (x)g(x)dx.
Lemma 2.4. If {fn} is two-norm convergent to f in L1 in the sense of Vitali, then {fn} is norm convergent to f in L1.
Proof. Put g to be a function such that g(x) = 1, for every x ∈ [0, 1] in the definition. 2
However, the converse of Lemma 2.4 is not always true. We consider the sequence {fn}, where
fn(x) =
½ n2 , for 0 < x < 1n 0 , elsewhere,
for every n. The sequence {fn} is norm convergent in L1to a function f , where f (x) = 0, for every x ∈ [0, 1]. However, {fn} is not two-norm convergent to f in the sense of Vitali in L1.
Based on two-norm convergence in the sense of Vitali in L1, we define two-norm continuous functional and shall prove Theorem 2.5 and Lemma 2.6 to have the repre-sentation theorem in Theorem 2.7.
A functional F defined on L1is said to be two-norm continuous in the sense of Vitali in L1, if
F (fn) → F (f ) as n → ∞
whenever {fn} is two-norm convergent in the sense of Vitali to f in L1.
Theorem 2.5. If g is essentially bounded on [0, 1] and F (f ) =
Z 1
0
f (x)g(x)dx for f ∈ L1,
then F defines a two-norm continuous in the sense of Vitali and linear functional on L1.
Proof. The linearity of F comes from the properties of the integral. The conti-nuity of F comes from the definition of two-norm convergence in the sense of Vitali in L1. 2
Lemma 2.6. Let F be a two-norm continuous in the sense of Vitali linear functional on L1. If G(t) = F (γt) for t ∈ [0, 1] then G is absolutely continuous.
Proof. Suppose G is not absolutely continuous on [0, 1]. Then there is ² > 0 such that for every δ there exists a partial division D = {[u, v]} satisfying
(D)X
|v − u| < δ and |(D)X
{G(v) − G(u)}| ≥ ².
For each n, take δ = n1 and D = Dn. For every x ∈ [0, 1], put fn = (Dn)X
|γv− γu|.
Then fn∈ L1 for all n and for every essentially bounded function g on [0, 1]
Z 1
0
|fn(x)g(x)|dx ≤ (Dn)X
M |v − u| ↓ 0 as n → ∞, where M is the essentially bound of g on [0, 1]. We define Fn(x) =Rx
0 fn(t) dt, for every n, we have {Fn} is uniformly absolutely continuous on [0, 1]. That is, {fn} is two-norm convergent to zero function in L1in the sense of Vitali. Yet we have
F (fn) = |(Dn)X
{G(v) − G(u)}| ≥ ² for all n.
It contradicts the fact that F is two-norm continuous in the sense of Vitali. Hence G is absolutely continuous on [0, 1]. 2
Theorem 2.7. If F is a two-norm continuous in the sense of Vitali and linear func-tional on L1 then there is an essentially bounded function g such that
F (f ) = Z 1
0
f (x)g(x)dx for f ∈ L1.
Proof. In view of Lemma 2.6 and using notation there, we obtain F (γt) = G(t) =
Z t
0
g = Z 1
0
γtg,
where g = G0 almost everywhere on [0, 1]. Since F is linear, for any step function f F (f ) =
Z 1
0
f g.
Take f ∈ L1, there is a sequence {fn} of step functions two-norm converges to f ∈ L1
in the sense of Vitali. Hence the general case of the theorem follows from the definition of two-norm continuity of F and the Dominated Convergence Theorem. 2
In conclusion, we have that every two-norm convergence sequence of functions on [0, 1] in the sense of Vitali in L1is norm convergence in L1 (Lemma 2.4). As corollary, every continuous functional on L1is two-norm continuous in the sense of Vitali on [0, 1].
Furthermore, we have proved the representation theorem on L1in the sense of Vitali in Theorem 2.7.
Acknowledgement. I would like to express my deepest gratitude to Prof. Lee Peng Yee from MME, NIE-NTU, Singapore for the valuable idea and suggestion of the re-search.
References
[1] Ambrosio, L., Fusco, NN., and Pallara, D., 2000, Functions of Bouded Variation and Free Dis-continuity Problems, Clarendon Press, Oxford.
[2] Hildebrandt, T.H., 1966, Linear Continuous Functionals on the Space (BV ) with weak topologies, Proc. Amer. Math. Soc. 17 (1966), 658 - 664.
[3] Khaing, K. Y., 2002, The Duals of Some Banach Spaces, Ph.D Thesis, Nanyang Technological University, Singapore.
[4] Indrati, Ch. R., 2009, Two-Norm Continuous Functionals on L∞, Journal of the Indonesian Society (JIMS), Vol 15 No. 2, October 2009.
[5] Khaing, K. Y., Lee P.Y., 2004, Orthogonally Additive Functionals on BV , Math. Bohemica 129, 411 - 419.
[6] Lee P.Y., 1989, Lanzhou Lectures on Henstock Integration, World Scientific.
[7] Lee P.Y., 1998, Teaching Calculus without ²δ. Matim´yas Matematika 21(1998) No. 3, 34 - 39.
[8] Lee P.Y. and V´yborn´y, R., 2000, Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press.
[9] Orlicz, W., 1992, Linear Functional Analysis, World Scientific.
[10] Riesz, F., Sz-Nagy, B., 1968, Le¸cons D0Analyse Fonctionelle, Gauthier-Villars.
[11] Rudin, W., 1976, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, Kogakusha, Ltd.
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