J.
Indones.
Math.
Soc.
(MIHMI)
Vol.
No.
, pp.
91Ae96.
TWO-NORM CONTINUOUS
FUNCTIONALS ON LO
Ch.
Rini Indrati Abstract.
In this paper we regard LO as a two-norm space and prove a representation theorem for two-norm continuous functionals defined on LO .
INTRODUCTION
The classical Riesz representation theorems is well-known.
It shows that every continuous linear functional F defined on the space C.
, .
of all continuous functions on .
, .
can be expressed in terms of a Riemann-Stieltjes integral.
That is, if F is a continuous linear functional on C.
, .
, then there is a function g of bounded variation on .
, .
such that Z 1 F .
) = f .
for f OO C.
, .
Let BV .
, .
denote the space of all functions of bounded variation on .
, .
In the language of functional analysis, the Riesz theorem says that the Banach dual of C.
, .
is BV .
, .
However, the Banach dual of BV .
, .
is not C.
, .
if we endorse BV .
, .
with the usual norm, namely, .
| V .
, .
) where V .
, .
denotes the total variation of f on .
, .
The main difficulty is that BV .
, .
is non-separable.
Hence the usual technique of proving such representation theorem no longer applies.
More precisely, the proof often contains the following two steps.
First, we prove the representation for some elementary functions, for example, step Secondly, we approximate a general function by a sequence of elementary Thus the representation for general functions follows from a convergence Received 21-09-2009.
Accepted 10-12-2009.
2000 Mathematics Subject Classification: 26A39 Key words and Phrases: functionals, two-norm LO space, essentially bounded functions.
Ch.
Indrati theorem for the integral.
If the space is non-separable, the second step does not Hildebrandt .
proved a representation theorem for BV .
, .
by regarding BV .
, .
as a two-norm space .
Here we state the theorem without proof in the form as given by Khaing .
A functional F defined on BV .
, .
is said to be two-norm continuous if F .
n ) Ie F .
) as n Ie O whenever V .
, .
) O M for all n and kfn Oe f k Ie 0 as n Ie O, where kf k = sup0OxO1 .
Theorem 1 If F is a two-norm continuous linear functional on BV .
, .
then there exist bounded functions f1 and f2 such that the following Henstock-Stieltjes integral and infinite series exist and Z b f1 dg O F .
= .
i ) Oe g O .
i )]f2 .
i ) for every g OO BV .
, .
, where ti , i = 1, 2, .
, are the discontinuity points of g, and g O is the normalized function of g.
In fact, the converse of Theorem 1 holds .
In this paper, following the same idea as above we regard LO as a two-norm space and prove a representation theorem for two-norm continuous linear functionals on LO .
Here LO denotes the space of all essentially bounded functions on .
, .
LINEAR FUNCTIONALS ON LO
A function F is essentially bounded if it is bounded almost everywhere.
Let LO be the space of all essentially bounded functions on .
, .
The two norms defined on LO , as suggested by Orlicz .
, are the essential bound kf kO and .
A sequence .
n } of functions is said to be two-norm convergent to f in LO if kfn kO O M for all n and Z 1 .
Oe f .
x Ie 0, as n Ie O.
A functional F defined on LO is said to be two-norm continuous if F .
n ) Ie F .
) as n Ie O whenever .
n } is two-norm convergent to f in LO .
We state without proof the big Sandwich Lemma .
We need it in proving a convergence theorem for the Lebesgue integral.
Two-Norm Continuous Lemma 2 If 0 O an O bkn for all n, k and lim lim bkn = 0 kIeO nIeO then limnIeO an = 0.
In what follows, when we say absolutely integrable we mean Lebesgue Lemma 3 If f is integrable and there is a positive constant K such that Z 1
Z 1
O K .
, for every bounded measurable functions g on .
, .
, then f OO LO and kf kO O K.
Lemma 4 If .
n } is two-norm convergent to f in LO , then f OO LO .
Proof.
Let An = fn , for every n.
Z 1
|An Oe Am | O
Z 1
fn Oe
Z 1
Z 1
fm | O .
n Oe f | .
n Oe fm | Oe01 .
Oe fm | Ie 0, as n, m Ie O.
That means {An } is Cauchy in real system, there exists a real number A such that An Ie A as n Ie O.
Let A > 0 be given.
There is a positive integer no such that for every positive integer n, n Ou no , |An Oe A| < A.
Therefore.
Z 1
|A Oe
Z 1
f | O |A Oe Ano | |Ano Oe
Z 1
fno | | Z 1 fno Oe f | < 3A.
That is, f is integrable.
If g is bounded measurable function on .
, .
, then g is integrable on .
, .
, then Z 1
Z 1
Z 1
Z 1
O |.
Oe fno ).
O |.
Oe fno ).
gk1 kfno kO kgk1 .
By.
Lemma 3, f OO LO .
Theorem 5 Let g be absolutely integrable on .
, .
If .
n } is two-norm convergent to f in LO then nIeO fn .
dx = f .
Ch.
Indrati Proof.
Since g is absolutely integrable on .
, .
, there is a sequence .
k } of essentially bounded functions on .
, .
such that Z 1 .
Oe g.
x Ie 0 as k Ie O.
Since .
n } is two-norm convergent, we have kfn kO O M for all n.
Hence the convergence of the integrals follows from Lemma 2 and the inequality Z 1
Z 1
Z 1
dx Oe f .
O 2M .
Oe g.
x Z 1 kgk kO .
Oe f .
Corollary 6 If g is absolutely integrable on .
, .
and Z 1 F .
) = f .
dx for f OO LO , then F defines a two-norm continuous linear functional on LO .
We define t .
= untuk 0 O x < t untuk t O x O 1.
A function G defined on .
, .
is said to be absolutely continuous if for every A > 0 there is > 0 such that |(D) {G.
Oe G.
}| < A whenever (D) .
Oe .
< , where D = {.
, .
} denotes a partial division of .
, .
in which .
, .
stands for a typical interval in the partial division.
We are using the notation of Henstock integral (.
, .
Lemma 7 Let F be a two-norm continuous linear functional on LO .
If G.
= F .
) for t OO .
, .
then G is absolutely continuous.
Proof.
Suppose G is not absolutely continuous on .
, .
Then there is A > 0 such that for every there exists a partial division D = {.
, .
} satisfying (D) .
Oe .
< and |(D) {G.
Oe G.
}| Ou A.
For each n, take P = n and Dn = D.
For every x OO .
, .
, there is .
, .
OO D, put fn .
= (Dn ) .
Oe u |.
Then kfn kO O 1 for all n and Z 1 .
x = (Dn ) .
Oe .
Ie 0 as n Ie O.
Two-Norm Continuous That is, .
n } is two-norm convergent to 0 in LO .
Yet we have F .
n ) = |(Dn ) {G.
Oe G.
}| Ou A for all n.
It contradicts the fact that F is two-norm continuous.
Hence G is absolutely continuous on .
, .
Theorem 8 If F is a two-norm continuous linear functional on LO then there is an absolutely integrable function g such that Z 1 F .
) = f .
dx for f OO LO .
Proof.
In view of Lemma 7 and using notation there, we obtain Z 1 F .
) = G.
= t .
where the integral is the Riemann-Stieltjes integral and G is absolutely continuous on .
, .
Note that G.
= 0.
Since F is linear.
Z 1 F .
) = f .
for any step function f .
Next, write g.
= G0 .
almost everywhere in .
, .
In view of integration by substitution .
74 Exercise 2.
20, we have Z 1 F .
) = f .
dx for any step function f .
Take f OO LO .
Then there is a sequence .
n } of step functions two-norm convergent to f .
Hence the general case of the theorem follows from Theorem 5.
CONCLUDING REMARKS
In conclusion, we have characterized completely two-norm continuous linear functionals on LO .
Acknowledgement The writer would like to express her gratitude to Prof.
Lee Peng Yee for the idea of the research.
Ch.
Indrati
REFERENCES