ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATY.CZNE XXX (1990)
Ro sel i Fe r n a n d e z (Sâo Paulo)
Characterization of the dual of an Orlicz space
Our objective is to characterize the dual of an Orlicz space & A{ X , E, fi), with the only hypothesis that (X , I , ц) is a measure space with no atoms of infinite measure.
This work originates from the reading of [12] and [13] of M. M. Rao, where we found some statements which were unclear to us. In particular the characterization in [13] seems to be incomplete, possibly for some fault in the fundamental definition. Here we present another characterization.
1. Preliminaries. If ф is a nondecreasing function from [0, co[ to [0, oo], such that 0 < il/(t0) < oo for some t0 Ф 0, the function A defined on [0, oo[ by the equality A(u) = jo ÿ{t)dt is called a Young function.
(1.1) R em ark. Let A be a Young function. Then (i) A is nondecreasing and convex;
(ii) the right derivative A' of A exists, is nondecreasing and is finite-valued on [0, b[, where b is as in (iv) below;
(iii) the function A defined on [0, oo[ as
is a Young function, called the complementary function of A;
(iv) uv ^ A(u) + A(v) for u, ve [0, oo[, with equality holding if and only if at least one of the relations v = p(u) or и = q(v) is satisfied by u and v, where
In what follows А, Я, b and b will be as in (1.1), and we will write a = sup (ne [0, oo[: A(u) = 0}, â = sup {ue [0, oo[: A(u) = 0}.
Moreover, (X, Z, p) will be a fixed but arbitrary measure space with no atoms of infinite measure (i.e. if F e Z and p(E) = oo, then there exists F eZ, F a E, such that 0 < p(F) < oo). Unless otherwise stated, our functions will be from X to R, and we shall employ the conventions that 0/0 = l/oo = 0 - oo = 0 and inf 0 = oo.
A(u) = sup {uv — A(v): p g[0, oo[},
b = inf (ug [0, oo[: Л(ы) = oo} and В = inf {ue [0, oo[: A(u) = oo}.
70 R. Fe r nan de z
The Orlicz space A is the space of all measurable functions f such that
§x A(k[f\)dn < со, for some /се]0, oo[. This is a complete space with the seminorm Ц-Ц^ defined by
\\f\\A = inf {/ce] 0, oo[: J A(\f\/k)dp < 1}.
x
(1.2) Pr o p o s i t i o n, (i) I f E e l and £,Ee££A, then \\ÇE\\A ^ bp(E);
(ii) for E e l the relation p(E) < oo implies that ÇEeJ?A, the converse holding if a = 0; moreover, if p(E) = oo and a > 0, then \\ÇE\\A = 1/a;
(in) for f e ^ A one has §x A(\f\)dp ^ 1 if and оп^У if \\f\W ^ (iv) if <5e[0, oo[ and /ejz?^, then
A(S)-ix{{xeX: |/(x)| > <5 l l / U H 1 ; (v) if f e <fA, then 1!/||л \ x Ai\f\)dn+ 1.
P ro o f, (i) follows immediately from A(u)/u for u e ]0 , oo[, and for this, observe that A(u)/u is nondecreasing.
To prove (iv), use (1.1 .i) and (iii). For the remaining assertions, see [2]. ■ (1.3) Pr o p o s i t i o n. For f in 3?A there is a sequence (s„) of simple functions in S£a such that
(i) 0 ^ s„ ^ sn + u for all n e N , if f ^ 0;
(ii) If-s„\ < | / | and |s„| ^ \f\ for all ne N;
(iii) (s„) converges to f ;
(iv) (s„) converges in p-measure to / ;
(v) if /се]0, oo[ is such that A ( \ f \ / k ) e t h e n lim J A ( \ f - s n\/k)dp = 0.
oo x
P ro o f. Discarding the trivial case, let \\ДА Ф0. The measurability of / guarantees the existence of a sequence (s„) of simple functions satisfying
(i)—(iii). Moreover, if E e l is such that/ is bounded on E, then (s„ÇE) converges uniformly to fÇE. From (ii) it follows that each sne ^ A.
To establish (v) it suffices to use (ii) and (iii) and apply Lebesgue’s Convergence Theorem.
It remains to prove (iv), and we first consider the case in which a > 0. Let B = {xeZ : \f(x)\ >2a\\f\\A}. Clearly (snÇBC) converges uniformly to /£ BC. By' (1.2.iv) we have p(B) < oo and hence, by Egorov’s Theorem, (s„£B) converges to f £ B in /r-measure. So (iv) holds in this case.
If a = 0, let c e ]0 , o o[ be such that 0 < A(c) < oo. Replacing к by \\f\\A in (v) we have
lim $ A ( \ f - s n\/\\f\\A)dp = 0,
n-+ oo x
and so there is a subsequence (s„J such that
f A ( \f -s J I \\ f\ \A)iii < 2 - t A(c/2k).
X
From this we see that p({xeX\ |/(x) —s„k(x)| > c\\f\\A/2k}) < \/2k for all k e N , and so (s„J converges to / in /^-measure. Replace (s„) by (snJ. ■
2. Characterization of the dual space (J?A)*. Denote by JtA the closed subspace of f£A spanned by the simple functions.
In [13] the author gives a representation for x*e(JtA)* as an integral, in the sense of Dunford and Schwartz [3], relative to a measure G defined as G(E) = x*{ÇE). Clearly this defines G(X) only if ^x Ef£A. Since this does not occur when a = 0 and p(X) = oo, the measure G is not defined on I in this case, and so we cannot apply the theory of [3]. Thus in [5] we introduce the concept of integration relative to a measure defined on an ideal.
To facilitate the reading, we transcribe from [5] some definitions and one proposition. For this, let sé be an algebra of subsets of X, and Ж an ideal of s4 (i.e. Ж is a ring, and for Ее Ж and F е Ж one has E n F e Ж). Moreover, let G be an extended real-valued finitely additive measure defined on Ж , and let v(G, •) be the total variation of G, which is obtained by replacing I by Ж in III. 1.4 of [3].
(2.1) De f in it io n. A function s is Ж -simple if there is a pairwise disjoint, finite sequence (Elt E2, ..., En) in Ж and there exists (cl5 c2, ..., cn)e R n such that s = ^ " =iC ^ £.. If v(G, (x e X : s(x) = cj) < oo for ct ф 0, we shall say that s is G-integrable and for E e s f we define
П
\sdG = £ CiG{E n Et).
E i = 1
(2.2) De f in it io n. A function / is G-integrable if there is a sequence (sj of G-integrable simple functions such that
(i) (sn) converges in G-measure to / , i.e. for every <5e]0, oo[ one has lim (inf{w(G, F): Р е Ж and {xeX : |/„(x)-/(x)| > 0} c F}) = 0;
n~> <x>
(ii) lim J \sn — sm\dv(G, •) = 0.
m, n~* oo x
If / is G-integrable and F e s / we define j / dG = lim j sndG.
E n ~* со E
(2.3) Pr o p o s it io n (Vitali Convergence Theorem). Let f be a function and (/„) a sequence of G-integrable functions such that
(i) {fn) converges in G-measure to f ;
(ii) \imviG'E)^ 0\E\fn\dv{G, •) = 0 uniformly in neiV;
(iii) for each e e ]0 , oo[ there is a set Fe in Ж with v(G, FJ < oo and such that j If„\dv(G, •) < e, for all ne N.
FÎ
Then f is G-integrable and for all Е е Ж one has S f d G = lim J f J G .
E n_>0° £
Next we take up our characterization of {JtA)*.
72 R. Fe r na n de z
From here on we agree that sé = Z and Z 1 = { E e l : ц(Е) < oo};
moreover, if a — 0 we set Ж — Z t , and if a > О, Ж = Z.
(2.4) De f in i t i o n. We shall say that Ge&a(X, Z, p , Ж) if G is a real
valued, finitely additive measure defined on Ж such that (i) G « in, i.e. if Е е Z and p(E) = 0, then G(E) = 0;
(ii) 0 ^ v(G, E) < oo for all Е е Ж ;
(iii) <Xq = inf{fce]0, oo[: IA(G/k, X) ^ 1} < oo, where
I-A(G/k, E) = sup
j
£ A ( r ~ § f ) ME,): ( £ „ £ 2...E J e â f t ,£)j,
for E e Z , and @(Zl} E) is the set of all pairwise disjoint finite sequences (E1, E2, ..., En) in Z t such that (J"= 1 E} c= E.
Unless otherwise stated, G will denote a real-valued, finitely additive measure defined on Ж such that G « /л.
(2.5) R em ark s, (i) For /се]0, oo[, the function IA(G/k, •) is a real-valued finitely additive measure defined on I .
(ii) If 0 < ocq < oo, then Ia( G /o l g, X) ^ 1.
(iii) If a g = 0, then G(E) = 0 for all E e Z l .
(iv) The function a<?> is a seminorm on the vector space &A(X, Z, p, Ж), and is a norm if Ж = Z l (i.e. a = 0).
(v) If G e ^ A(X, Z, p, Ж), then by (1.2.Ü) and (2.4.ii), every simple function in A is Jf’-simple and G-integrable.
In (2.8) below, for a fixed / e E£A we shall find a sequence (s„) of simple functions in S£a such that
\ f d G = lim J sndG for every E e Z and all Ge&a(X, Z, p, Ж).
E n~>co E
First, some propositions.
(2.6) Pr o p o s i t i o n. I f b = oo and G is such that olq < oo, then G is p-continuous, i.e. И т^(Л)_ 0 G (A) = 0.
P ro o f. See Lemma 6 in [12]. ■
(2.7) Pr o p o s i t i o n. Let Ge&a( X , Z, p, Ж). I f seL£A is a simple non
negative function, then
J sdv{G, •) ^ c\\s\\A,
x
where с = 2а£ if a — 0 or p(X) < oo, and c — 2aG + av(G, X) if a > 0 and p(X) = o o .
P ro o f. Eliminating the trivial case, suppose j|sj|^ / 0.
Let s = Yj = ^ ci^Ei be as in (2.1) with ct > 0.
First we observe that if (F1} F2, ..., Ft) is a pairwise disjoint finite sequence in Z 1 and # 0, then by (l.l.iv) we have
(1) X С|В Д | ^ а § И л j =i
M L
Let J = {ieN: 1 ^ i < n and Eie Z 1}. Using (1.2. iii), (2.5. i), (2.5. ii), (1) and observing (2.5. iii) it follows that
(2) Y ,c A G ,E ^ 2 a .l\\s \\A.
ie J
By (1.2.Ü) we obtain
(3) £,) = а ^ с М в Л л ^ С , E,) « a||sLi>(G, X).
i$J i$J
From (2), (3) and (1.2. ii) we obtain the desired result. ■
(2.8) Th e o r e m. For f in f£A there is a sequence (s„) of simple functions in f£A satisfying (1.3.i), (1.3.ii), (1.3.iii) and such that for all Ge^a{X, Z, p, Ж) one has:
(i) (s„) converges in G-measure to /;
(ii) / is G-integrable and
I f dG = lim j s ndG, for Е е Z.
e " - >o° E
/Is a consequence, the following hold:
(iii) \\x fdG\ < с ||/ ||А, with c as in (2.7);
(iv) )xfd(yG1 + G2) = y$xfdG1 + Sx fdG2, for y e R and Gt , G2e
* A X , z , ii, * ) •
P roof. To establish (i) and (ii), it suffices to consider / ^ 0 .
If b — oo, let (s„) be as in (1.3). From (2.6) we deduce that (s„) converges in G-measure to / .
If b < oo, we have |/ | ^ 2b \\f\\A /i-a.e. by (1.2.iv) and, since/is measurable, we can take a sequence (s„) of simple functions satisfying (1.3.i), (1.3.iii) and converging uniformly to / except on a set F e F with p(E) — 0. Clearly sne л for n e N, v(G, ' ) « p and (s„) converges in G-measure to /.
Now we will prove that the sequence (sn) above also satisfies (2.3.ii) and (2.3. iii), and thus obtain (ii).
For all n e N , let yn = §x sndv(G, •). Since (y„) is bounded by (2.7), and nondecreasing, given s e ]0, oo[ there is an n0e N such that
$sndv(G, ■) = j sndv(G, • ) - J sndv(G, •) < e/2 + J snodv{G, •),
E X Ec E
for all n ^ n0 and E e Z .
If M — 1+max {|s„0(x)|: x e l } , ô = e/(2M) and Fe = ( x e l : s„0(x) Ф 0}, we have F£e J a n d (s„) satisfies (2.3.ii) and (2.3.iii).
74 R. F e r n a n d e z
For (iii), use (ii), (2.7) and note that
|Js„dG| ^ J|sJd » (G .-), for n eN .
X X
To establish (iv) it suffices to use (ii). ■
Our next objective is to define a norm on УА(Х, E, g, Ж) and to prove that this space is isometrically isomorphic to { J if f.
(2.9) Pr o p o s i t i o n. The function defined by
|| G ||л = sup{|J/dG |: feS eA and ||/||л < 1}
x
is a norm on @A(X, E, g, Ж). Moreover, one has
IIGh = sup {|f fdG\: f e J t A and \\f\\A ^ 1}.
x
P ro o f. The first assertion follows trivially from (2.8.iii) and (2.8.iv). For the second, observe that simple functions in <£A belong to JiA and use (2.8. ii). ■ (2.10) Pr o p o s i t i o n. I f Gg^a(X, E, g, Ж), then the function x* defined on MA{_ie^\ by x*(f) = \ x fdG, belongs to { J i f f [(Jzf*)*] and ||x*|| = ||G||^.
(2.11) Pr o p o s i t i o n. Let x* G{Mff and let G be the function defined on Ж by G{E) = x*{ÇE). Then Ge&A(X, E, g, Ж) and
(i) x*(f) = fx f d G for all f e M A, and x* = ||G||^.
P ro o f. We may suppose ||x*|| # 0 . It is clear that G is a real-valued, finitely additive measure, and G « g.
To prove that v(G, E) < oo for all E g Ж , it is enough to observe that if
E g Ж and (Et , E2, . En) is a pairwise disjoint finite sequence in Ж with и ?=1^г ^ E then taking J — (ieiV: 1 ^ i ^ n and x*(£Êi) ^ 0}, we have
î |G(£,)I = x*(« v u e ) ^ 2 Иx*|| I ie jx.
i = 1 i eJ i $J
Next we will prove that < ||x*|| < oo.
Let c e ]0 , 1[ and let (£ ls E2, ..., En) be an arbitrary pairwise disjoint finite sequence in Г 1. If щ — c|G(Ei)|/(||x*||jit(£j)) belongs to [0, (1.2.i),
c { = limA'(t) for ie { l, 2, ..., n ] and / = £ C i S g a ( G ( E ^ E t i
i= 1
then and by (l.l.iv) and (1.2.v) we have
> c|x*(/)|
i= z1
A{c^ + A c\G(EÙ\ ' х *\\ дЩ
» II/IL - 1 + Z Л| №
i = 1 1**11
From this relation it is easy to see that aq ^ ||x*||/c, and since c e ]0 , 1[ is arbitrary, we conclude that a§ ^ ||x*||. Thus G e ^ ( l , Z, p, Ж).
To obtain (i) it is enough to observe that (2.10) is true and that x*(s) = Jx sdG, where s is a simple function in &Л- ■
From (2.10), (2.11) and (2.8.iv) we obtain in (2.12) below a characterization of {Jl'A)*; it will follow that &A{X, Z, pi, Ж) is a Banach space.
(2.12) Theorem. There exists an isometric isomorphism of { J i f f onto УА(Х, Z, p, Ж), given by the mapping x*i—>G where G(E) = x*(££)for ЕеЖ, and the following holds:
(i) x*(f) = \ x fdG for all f e J i A and ||x*|| = ||G||*.
(2.13) Comment. In [13], with a given х*е{Ла)* the author associates a measure G defined on Z, satisfying some conditions. To obtain (2.12) it was necessary to replace the domain of G by Ж , and condition (i) of Definition 3 in [13] by (2.4. ii).
3. Characterization of the dual space (ЖА)*. Let JTA = .J}A \JtA, write / ~ f \ M A for a n y / e i ^ , and denote by d(-) the quotient norm on JfA. One
can easily show that
d(f) = inf {||/+s||^: s is a simple function in A).
(3.1) Definition. Let / , де<ЖА. We write / < g if there exist f e f and gxE§
such that f x ^ gx.
In [13] it is proved that the above relation is a partial ordering on JfA and that { J f f f is a vector lattice. Two results which are used in the proofs, and are not obvious to us, are established in (3.4) and (3.5). First, two propositions.
(3.2) Proposition. Let a > 0 or p(X) < oo. I f f is a measurable function, bounded p-a.e, t h e n f e J i A.
P ro o f. We may suppose / to be bounded. Since there exist x e ]0 , oo[
such that A(x) < \/(p{X)+ 1), and a sequence (s„) of simple functions, conver
ging uniformly to /, it is easy to verify that lim ,,^ ||/ — s„||x = 0. From a > 0 or p{X) < oo it follows that / , sne f£A for all n e N . Thus f e J i A. ■
(3.3) Proposition. Let
f A = { f e & A: { A(k\f\)dp < oo for all /се]0, oo[}.
x
Then f A is a closed subspace of JiA and (i) f A a M A cz <£a,
(ii) if a = 0 and b = oo, or if p(X) < oo and b — oo, then f A ~ JiA\
(iii) if a > 0 and b < oo, or if p(X) < oo and b < oo, then f£A = JiA.
P ro o f. From (1.3.v) it follows easily that The remaining assertions are proved in Lemma 2 in [13]. ■
76 R. F e r na n de z
(3.4) Pr o p o s i t i o n. I f g s J t A and f is a measurable function such that l/l ^ \g\ g-a.e, then f t J ( A.
P ro o f. We may suppose | / | ^ \g\. Let (s„) be a sequence of simple functions in S£A such that lim,,-^ \\g — s j ^ = 0.
First we observe that for /се]0, oo[, if n e N is such that 2\\g — sn\\A ^ k, then by (l.l.i) and (1.2.iii), for all H e l we have
(1) f A{\f\/k)dg ^ f A(\g\/k)dg ^ i + i j A(2\s„\/k)dg.
H H H
Suppose b = oo and a > 0. From lim,,-,^ Il g — = 0 there exists an n0e N such that \\g — ^ 1.
Let B = {x e X : \g(x) — sno(x)\ ^ 2a}. Then g(B) < oo (1.2.iv), and the inequalities |/ | ^ \g\ ^ \g — s„0| + |s„0|, together with (3.2), imply that fÇBceJtA.
Moreover, it follows from (1) that f£,Bz f A <= Л А. So f ^ M A in this case.
Let b < o о, a — 0 and En = {xeX : s„(x) ф 0} for neiV. Then l/l ^ 2b\\f\\A g-a.e. (1.2.iv), g(En) < oo (1.2.Ü) and f ^ Ene J I A (3.2), for all ne N.
Using (1) we obtain lim,,-,.,» ||/—№е„\\а = 0, and since JtA is closed in ££A, we conclude that / e MA also in this case.
The remaining cases are easy (3.3). ■
(3.5) Pr o p o s i t i o n. I f f , geJ?A and \f\ ^ \g\ g-a.e, then d{f) ^ d(g).
Pro.of. We may suppose | / | ^ \g\ and, by (3.4), that д ф ^ А.
It suffices to prove that for all e e]0 , oo[ we have
(1) d ( f ) ^ d ( g ) + s.
For a fixed e e ]0 , c o [ , let s eJ£a be a simple function such that
0 < d(g) ^ Hgf + sll^ < d(g) + s and let = { x e l : s(x) # 0}.
As дфЛ А we have b < oo and a = 0, or b = oo (3.3.iii).
If b < oo and a = 0, then giHf) < oo (1.2.Ü) and |/ | ^ 2b\\f\\A g-a.e.
(1.2.iv). Since f £ HlEJfA (3.2) and
Шн\\\а < Ы щ\\а < \\(9 + s)^Hc\\a < d(g) + £, we conclude (1).
For the remaining cases, we first note that if HeZ and if we have A(|s|/((a — l)\\g + s\\f))ÇHE & 1 for all <xe]l , oo[, then
(2) d ( ( f ( BT ) K d ( g ) + s .
In fact, by (l.l.i) and (1.2.iii), I -I ( , 11 T ) d l , - s i л ( lgl )
h \a|l 9 + s\\aJ h \<x\\9 + s\\aJ
, 1 a —1 c
dg ^ -H---j A N dg,
sa|l 0 + s \ \ J h \ot\\g + s\\AJ a « я \(а-1)110 + «1и, and hence A(\f\/(<x\\g + s\\a))Çh eJ ? l . From (1.3) there exists a simple function
s0e^ a such that | | ( / -s0KhIIa ^ a Il0 + «L- Thus
d((f£HT) < \ \ / - Зо)£н\\а ^ <x\\9 + s\\a < a(d(g) + e) and, since a s ] l , oo[ is arbitrary, (2) holds.
If b = oo and a = 0, then < oo (1.2.Ü) and we obtain (1) replacing H by X in (2).
If b — oo and a > 0, let H2 = {x e X : \g{x) +s{x)\ ^ 2a\\g+ s\\A}. Then (1.2.iv) tells us that g(H2) < oo and we may replace H by H2 in (2). Moreover, it follows from |/1 ^ \g\ ^ |0 + s| + |s| that f£ Hc is bounded, and thus ft,Hc2E.ÆA (3.2). Therefore (1) holds. ■
Let £ be a pseudonormed vector lattice, x e E and z* eE*. Then we will set as usual x + = x v 0, x_ = ( — x) v 0 and |x| = x + + x _ . Also, we will write z* ^ 0 if z*(y) ^ 0 whenever y ^ 0.
(3.6) Pr o p o s it io n, (i) J f A is a vector lattice, and if / , §еЛГа, then / v g = (max {/, g}J, f л g = (min {/, g}J and ( | / / = 1/ 1;
(ii) J fA and {jVf f are Banach lattices;
(iii) if х*е(ЖА)* and x* ^ 0, then
||x*|| = sup(x*(/): / ^ 0 and j A(f)dpt < 00};
x
(iv) if x *é ( J^ A)*, then ||x*|| = ||(x*)+|| + ||(x*)_||;
(v) if x*, y*e(jVA)* and x*, y* ^ 0, then
llx* + y*ll = llx*ll + lly*ll;
(vi) if f, g e J f A and /, g ^ Ô, then d ( f v g) = max{d(f), d(g)}.
P ro o f. Except for (iii), these assertions are proved in [13]. Assertion (iii) follows from Lemma 6 in [13], upon observing that |/ | = .fl/l/for ■ From (3.6. ii) and (3.6. v), and from (3.6. ii) and (3.6.vi) we have, respectively, that (jVa)* is an L-space and that Jf A is an M-space.
From here on v will denote a real-valued, finitely additive bounded measure defined on I , v(v, •) the total variation of v, vt = (v + u(v, -))/2 and v2 = (a(v, •) —v)/2. Also we will denote by 0 the set of all pairwise disjoint finite sequences (El , E2, ..., En) in I such that (J"=1£ f = X.
(3.7) De f in i t i o n. L e t/e jS ^ . I f f is nonnegative, we define j f d v j = inf{ £ d((ftcy)vj(Ei: (£ 1; E2, .... E„)e3>},
X i = 1
f o r j6{ l ,2}.
In the general case we define
I fdv = ( J /+ dvx - J /+ dv2) - (J /_ dv1 - j /_ dv2).
X X X X X
78 R. F e r n a n d e z
(3.8) R em ark s, (i) From Lemma 10 in [13] and (3.5), one has I (yf+g)dv = y J f d v + §gdv, for y e R and
x x x
and
If f d v I ^ f | / | dv, + f l/l dv2 « d(\f\')(v, + v2)(X) = d ( f Mv, X), for /<= .
X X X
(ii) If x*(f) — J f d v , for f e j V A, then x * e ( J rA)* and ||x*|| ^ u(v, X).
x
(iii) If v is nonnegative, E e l and vE is defined on I by vE(F) — v(E nF), then j x f ^ E dv = \ x f d v E for all 0 ^ f e F F A.
(iv) If v and v are real-valued, finitely additive bounded measures defined on E , f e F £a and ye R , then
if d ( yv + v) = y \ f d v + \ f d v .
X X X
(V) If / e JTA a n d / ^ 0, there exists a nonnegative representative of /.
Next we take up our characterization of {JFff.
(3.9) De f in i t i o n. We shall say that ve'Y'jffF, I , pi) if v « pi and there exists f e J ^ A, f ^ 0 with d(f) ^ 1, for which the following holds:
(i) if E e l and v(v, E) ф 0, then d((fÇEf) = 1.
It is easy to see, using (3.6.vi), that Уд(Х, I , pi) is a vector space.
(3.10) Co m m e n t. In [13] (page 573), in place of i^A(X, I , pi), the author uses &A{fi), the space of all v « pi such that there exists / e J2^ with l x A{\f\)dpi < a o ,f 4 J ? A and the following holds:
(i) the support of v lies in the support o f/, where “support of v” is defined in [13] (page 571) as “the sets E for which v(v, E) > 0”.
The author claims that if v e & A{pi) is nonnegative and x*(/) = \ xfdv for a ll/e Jf A, then ||x*|| = v(X). To prove this, he considers л: = (Ex, E2, ...,£ „ ) e 0* with v(Et) > 0 and defines f = f x + / 2 + ••• +/„ where \ xA(\f\)dpi < oo, d(fy = 1 and support o f/ is Et, for ie{ 1 , 2 , . . . , n}. Then he writes: “Thus Y!i=i d((f%EiT)v(Ei) = v(X) and refining the partition n on the left yields x * ( f ) = v(X)”. This last assertion is not clear to us, for / depends on the partition 7i. However, we observe that if there is an / satisfying (3.9. i), / ^ Ô, then it is easy to see that x*(f) = v(X). Thus we have replaced, in our characterization, (3.10. i) by (3.9. i).
(3.11) No t a t i o n. If E e l and х*е(.Л/1)*, we will denote by x | the function defined on J fA by x E(f) = x*((f£Ey).
(3.12) Pr o p o s i t i o n. Let 0 ^ х*е(ЖА)*. Then there exists geJ?A, g ^ 0, with d(g) ^ 1, such that
(i) ||xl|| = x*((g£EY) for all E e l ;
(ii) if E e l and ||x||| ^ 0, then d((gÇEy) = 1.
Proof. By (3.6.iii), for n e N there exists a nonnegative/„eJS? 4 such that JxA(f„)dp < 00 and x*{fn) > ||x*|| — l/n. Moreover, from (1.3) we know that there is a nonnegative hnef£A such that \AA(hn)d/x < 1/2" and /j„ =/„.
Let h = lim„_>00(max{/i1, . . . , hn}) and observe that if E = {x eX:
h(x) = 00}, then p(E) = 0. In fact, if p(E) > 0, there exists F e l l , F <= E such that jx{F)> 0. By (1.1. iv) and (1.2. iii) we have
oo-ju(F) = j ÇF h d p ^ U f\\a lim
X k- + 00
H ï &)
dp + E J A(hn)djx < 00,п = 1 X
and so p(F) = 0, which is a contradiction.
Let g = h£Ec Then j x A(g)dp ^ sc 1, and since x*(g)^ x*(£n) = x*(fn) > \\x*\\ — l/n, for all neN, we conclude that
||x*|| = x*(g). So for EeE, using (3.6. v), we obtain
0 ^ llxlll ~ х * ( Ш ) = x*((gÇECT) - Hxtll ^ 0, and thus
llxlll = х * ( Ш ) < WxiWdfaZJ) ^ llxlll. .
(3.13) Th e o r e m. Let 0 ^ x* e (.Ж^)*. Then there exists a unique
v e'VA(X, E, fx), nonnegative, defined by v(E) = ||x |||, such that (i) x*{f) = $x fd v for all f e J f A, and ||x*|| = v(X).
P ro o f. Let v{E) = ||x||| for EeE. Then v e'Ÿ'A(X, I , p) ((3.6.v) and (3.12)). To show that (i) holds define z*(f) = $xfdv for f e J A A\ we shall prove that I)z* — x*I) = 0.
For Ô 4 and ж = (E1, E2, En)e0> we have
Î d((/^,r)v(£,.) = £ d((/{E,r) llxl.ll » £ **((/<W) = * * (Л
i = 1 i = 1 i = 1
and from this it is clear that z * ^ x * . Thus ||z*|| = ||x*||, for
||z*|| ^ v ( X ) = ||x*|| (3.8.ii), and also ||(z* — x*) + x*|| = ||z* — x*|| + ||x*||
(3.6.v). Hence ||z* — x*|| = 0 .
To prove the uniqueness of v suppose that v e'V A{X, E, p) is nonnegative and x*(f) = \ x f dv for all f c J T A. Then by (3.8.iii) and (3.8.i) we have
x u f ) = x*((fiEj) = и м и < d ( f ) m , x
for j > 0 and Ee E. So v —v is nonnegative, belongs to УА{Х ,Е ,р ) and i x f d { v - v ) = 0 for all f E f £ A (3.8.iv).
Since v - v e f ^ ( I , E, p), there exists Ô ^ g e/ 4 such that d((g^Ef) = 1 for EeE with (v — v)(£) # 0. Thus \xQd{v — v) = (v — v)(X), and hence (v-v)(X) = 0. ■
(3.14) Th e o r e m. There exists an isometric isomorphism of (JfA)* onto
^д(Х, E, p), given by the mapping x*i—>v, such that (i) x*(f) = Jx f d v for all f e J f A, and ||x*|| = v{v, X).
80 R. Fer nan de z
P ro o f. If х * е ( Ж А)* we will write y* = (x*)+, z* = (x*)_ and define, for all E e Z , v3(£) = ||y*|| and v4(£) = ||ze||. Let v = v3 —v4.
To prove (i), use (3.13) and (3.8. iv) to write x*(f) = Sfdv for all f e JTA,
x and (3.6.iv) and (3.8.ii) to obtain
v3(X) + v4(X )= №*11 + 11**11 = ||x*|| ^ v(v, X) ^ v3(X) + vA(X).
To show that the mapping х*ь->v is onto, consider x e ' V A{X, Z, g) and define x*(f) = $ x f d v for /еЛ~А. Then, writing vl = (v + u(v,-))/2 and v2 = (u(v, •) — v)/2, we have
j - J f dv2 = j f d v = x*(f) = \ f d v = \ f d v 3 - J f dvA,
X X X X X X
for all f e j T A, and thus
J /rf(Vi+v4) = J f d{y2 T v2) for all / e ^ 4-
X X
Since \ е У А{Х, Z, g), it is easy to verify that vl5 v2 6^ (X, Z, g), and thus vj + v4 = v3 + v2 (3.13). Hence v = v.
Finally, observing that x*i—>v has a linear inverse, we conclude that the mapping is linear. ■
It is immediate, by (3.14), that ЖА(Х, Z, g) is a Banach space.
4. Characterization of the dual space (£ТЛ)* . As in Section 2, set Ж = Z if a > 0, and Ж = Z x if a = 0.
(4.1) Th e o r e m. Let x* e ( ^ A)*. Then there exist a unique G e ^ A(X, Z, g, Ж), defined by G(E) = x*(ÇE) for Е е Ж , and a unique z* e{Jtf)1 such that
(i) x*(/) = j x f dG + z*(f), for all f e S T A.
P ro o f. The same as in Proposition 2 of [13], noting that (2.10), (2.11) and (2.1 2) hold. ■
(4.2) Pr o p o s i t i o n. Let x* e(JTA)*, G and z* as in (4.1), and y*(f) — \ x f d G
for all f e L£a. Then \
(i) if x* ^ 0, then y* ^ 0 and z* ^ 0; (ü) Il \y*\ + \z*\ II = 11/41 + 11**11;
(iii) |x*| = |y*| + |z*| and |y*| л |z*| = 0.
P ro o f. If x* ^ 0, it is clear that y* ^ 0. Thus for (i) it suffices to prove that x* ^ y*. For this let 0 ^ f e S T A and take (s„) as in (2.8). Since z*e{Æff- and x* ^ 0 we have
y*(f) = lim j sndG = lim x*(s„) ^ x*{f).
n~* 00 X 00
For (ii), let e e ]0 , oo[. Observing that if w*e(JTA)*, then ||w*|| = || |w*| ||
(page 239 in [7]), thus there exist f g e £TA, nonnegative, with \\f\\A ^ 1,
\\g\\A ^ 1 and such that
Ilk*!! < l.y*l(/) + £/2 and ||z*|| < \z*\{g) + e/2.
From (1.3), for k e N fixed there exists se.J/A with 0 ^ s ^ g and such that J A(g-s)dp < (1 + l/7c)-J A(f)dp.
X X
Let h = max {/, (g — s)]. Then hefTA and
f kh \ к к
\ A 7TT ^ r r r f ^ 7 Т т [1 + f ^ L
X \ k + 1 / k + i x k + l x x
Thus \\h\\A < (k+ l)/k.
Hence we have
l l / I K 11**11 < \y*\(f) + \z*\(g) + e = \y*\(f) + \z*\(g-s) + e
< \y*\(h) + \z*\{h) + £ = (|y*l + |z*|)(/i) + e
< Il \y*\ + \z*\ II НЛИл + е < II \У*\ + \г*\ ll((/c + l)/fc) + e.
Since s and к are arbitrary we conclude that (ii) holds.
(iii) may be found in [11] (page 40). ■
(4.3) Pr o p o s it io n. There is an isometric isomorphism of ( J i f f onto ( Л А)*
given by the mapping j : z * w h e r e x * is defined by x*(f) = z*(/).
In the next theorem we present our characterization of (J ’f * .
(4.4) Th e o r e m. There is an isometric isomorphism of (STA)* onto the Banach space УА(Х, Z, p, Ж) x ЖА(Х, I , p) given by the mapping x*h->(G, v) such that
(i) x*{f) = $x f dG + ] x fd v for all f e <£A\
(ii) 11**11 = \\G\\a + v(v,X),
the first integral being as in (2.2) and the second as in (3.7).
P ro o f. This is a consequence of (4.1), (4.3), (3.14) and (4.2), by observing that ||x*|| = || |x*| || for x*e(J>a)* [7]. ■
5. Concluding remarks. (I) The characterization of (f £ f f obtained by Andô in [1] goes through the spaces and J AjjiA with the hypotheses p(X) < oo and b = oo. From (3.3) we know that under these hypotheses the spaces / A and JiA coincide.
(II) Using (2.6), (2.8) and Radon-Nikodym’s Theorem, one can prove that if p(X) < oo and b = oo, given Ge c$a{X, I , p, Г Д there is g e f f A such that
(i) §xfdG = §x fgdp, for all f e J A,
(ii) \\G\\A = swp{$x \gh\dp: hef£A and ||й||л <1}.
Hence, in this case, Theorem (4.4) above tells us that given x*e(J?A)*, there exist gef£-A and ve r A( X , Z, p) such that
(iii) x*(f) = \xf9dp + ^xfdv, for all feS£A,
(iv) ||x*|| = sup{jx \gh\dp: h e f T A and \\h\\A ^ 1} + v(v, X).
Thus the result obtained by Andô in [1] and Theorem (4.4) coincide when h{X) < oo and b = oo.
(III) In [4] we also prove that (iii) and (iv) above still hold when (X, Z, p)
~ Commentationes Math. 30.1
82 R. F e r na n de z
is a (j-finite measure space and # A — JtA, or when à = 0 and f A = JiA.
(IV) If f A Ф {0} and # A Ф MA, there exists Ge&A(X, I , g, Z) such that for all the following assertion is false:
J f d G = J fgdu, for all f e J f A.
x x
This will be proved below.
From (IV) it follows that (iii) is not true if / А Ф {0} and J A Ф JiA.
P r o o f of (IV). By (3.3) we know that fi{X) = go, b = oo and a > 0.
Therefore E,x e J i A and %ХФ</А> also я = 0.
By Hahn-Banach’s Theorem and (2.12) there exist x*e (J^a)* and Ge&A(X, Z, ji, Г) such that
^*(^лг)#0 5 **(/) = 0 for all f e f A, x*(f) = j fdG, for all f e J t A.
x
Suppose there is a ge££A such that
{ fd G = \f g d g , for all f e M A.
X X
Let y e ]0 , oo[ and F = { x eX : \g(x)\ > y} be such that g(F) > 0 (this F exists because x*(^x) ф 0).
As à = 0, and A
we conclude that g{F) < oo, and so ^ s g n g e ^ . Thus
0 = x*(Çpsgng) = $(ÇFsgng)gdn = j ÇF\g\d[i ^ yg(F) > 0,
X X
which is impossible. ■
This work is part of our Master’s Dissertation written under the guidance of Dr. Iracema Martin Bund.
References
[1] T. A n d ô , Linear functionals on Orlicz spaces, Nieuw Arch. Wisk. 8 (3) (1960), 1-16.
[2] I. M. B u n d , Birnbaum-Orlicz spaces, IME-USP, Sâo Paulo, 1978 (Notas do Instituto de Matemâtica e Estatistica da Universidade de Sâo Paulo, Série Matemâtica, 4).
[3] N. S. D u n f o r d and J. T. S c h w a r tz , Linear Operators, part I : General theory, Interscience, New York 1967.
[4] R. F e r n a n d e z , Caracterizaçâo do dual de um espaço de Orlicz, Dissertaçâo (Mestrado), Instituto de Matemâtica e Estatistica da Universidade de Sâo Paulo, 1986.
[5] —, Integraçâo em relaçâo a medidas definidas em ideais, Trabalhos apresentados, 23°
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[6] H. H u d z ik , Orlicz spaces o f essentially bounded functions and Banach-Orlicz algebras, Arch.
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[7] J. L. K e lle y and I. N a m io k a , Linear Topological Spaces, Graduate Texts in Mathematics, 36, Springer, New York 1963.
[8] M. A. K r a s n o s e l’s k ii and Ya. B. R u tic k ii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen 1961.
[9] J. M u s ie la k , Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, New York 1983.
[10] W. O r lic z , On integral representability o f linear functionals over the space of (p-integrable functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 567-569.
[11] A. L. P e r e s s in i, Ordered Topological Vector Spaces, Harper&Row, New York 1967.
[12] M. M. R ao, Linear functionals on Orlicz spaces, Nieuw Arch. Wisk. 12(3) (1964), 77-98.
[13] —, Linear functionals on Orlicz spaces: general theory, Pacific J. Math. 25(3) (1968), 553-584.
INSTITUTO DE MATEMÂTICA E ESTATISTICA UNIVERSIDADE DE SAO PAULO
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