On bimodular spaces

A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

T. M. J ^

(Bydgoszcz) and J . M

(Poznan) On bimodular spaces

1. Let X be a real linear space. A functional

defined in X with, non-negative values including + oo, satisfying the following conditions :

A .l. q (0) = 0, A.2.

=

A.3. д(ах-\-(Зу) < ag (x) ( } q (y) for a ,/? > 0, a + / S = l ,

is called a convex pseudomodular in X. If a pseudomodular q satisfies additionally the condition

A.l'.

= 0 implies x = 0 it is called a convex modular in X.

The linear subspace

X~ = {x : q (2.x) -> 0 as A 0 -f}

of X is called a modular space. The functional INI? = inf je > 0 : q < 1 j

is a homogeneous pseudonorm in X~ and satisfies the condition

||a?||~ < 1 implies

^ N 1?

(see [2], [3]).

How, let X, Y be two real linear spaces, and let q ( x , y) be a functional defined on the Cartesian product X x Y with non-negative values in­

cluding + oo such that

1. for every fixed y* Y , g( x, y) is a pseudomodular in X, 2. for every fixed xeX, g( x, y) is a pseudomodular in Y, 3. for every real А, g(2.x,y) =

Xy).

Then q is called a bipseudomodular in X x Y . If additionally, g(x, y)

= 0 for all у e Y implies x = 0, then q is called a bimodular in X x Y . 1.1. Let q be a bipseudomodular in X x Y and let us denote by qv ( x ) the pseudomodular q ( x , y) in X with a fixed y e Y, and by gx(y) the pseudo­

modular g(x, y) in Y with a fixed xeX. Then x e X Qy if and only if y c Y ^ .

Proof. The condition x e X ey means that

{X

) 0 as A ^ 0 + . But

(X

) — д(Хх, у ) =

(

, Xy) = Qx(Xy)j hence

(X

) 0 as X -> 0 + if and only if Qx(Xy) -> 0 as X -> 0 + , i.e. if and only if y e Y Qx.

1.2. Let us denote by ( X x Y ) g the set of points { x , y ) e X x Y for which x e X ey, or equivalently the set of points ( x , y ) e X x Y for which y e Y ex. The set ( I x I ) 5 is called the bimodular set.

1.3. I f ( x , y ) € ( X x Y ) e, then \\x\\gV = \\у\\вх.

This statement follows from the equalities

\\

\\qV = inf je > 0 : < 1 j = inf je > 0 : < l j = M\Qx>

1.4. We define for (x, y) e{X x Y ) e, \\{x, y)\\g = \\x\\ey = \\y\\Qx.

The following properties are obvious:

i° m , y ) \ \ e = \\{x, o)iie = o ,

2° \\(Xx,y)\\e = \\(x,Xy)\\g = |A|||0»,y)lle,

3° + y)\\Q ®

y)||e + ||0»a, y)\\Q, Ц(<», y x + y z)\\e < ||(<», y1)||e + -Н10»,Уа)Ие,

4° if 110», 2 /)||e < 1, then q { x , y) < \\{x, y)\\e.

1.5. Let X® Y be the algebraic tensor product of the linear spaces X and T (see [1], p. 80), and let p be a bipseudomodular in X x Y . Let

П

(X®Y)e be the set of all elements {xfâyfja^Xt&Y such that there

•exists an element г=1

П П

£ » Vi) ai € X У** ai

г=1 г=1

for which 0»г, Уд€( Х X Y)e for i = 1 , 2 , n. It is obvious that (X®Y)e is a linear subspace of the linear space X® Y. The space (X® Y)„ will be called the bimodular space.

1.6. Every element 2 e(X®Y)e has a representation П

Z =

^4®&,

i —1

where {x^ yi) e { X x Y ) e for i = 1, 2 , . . . , n .

P roof. By the assumption, there are {xi , y i) e ( X x Y ) g and real П

numbers ai such that £ {xi , ÿf) • at ez.

i =

l

Taking xi = xi and yi — у we have z = ^ ]xi® yi and it is suf-

г= l

ficient to show that (xif у^е( Х x Y)e. But, by the assumption, Xyf) -> 0 as A- >0 + . Hence Q(xt , Xyf) = /Uqiq) -> 0 as A - ^ 0 + , i.e.

Уг)е{ХХ Y)e.

1.7. The functional

INIe = inf ПК, Уг)11в : y^ €Z

i = 1 г = 1

and (xi} yi) e ( X x Y ) Q for i = 1, 2 is a pseudonorm in the bimodular space (Z®Y)e.

Proof. Let в be the zero element of the tensor product then (O,O)c0, where ( 0 ,0 ) e ( X x Y)Q. Hence ||0||e < ||(0, 0)||e = 0 , i.e. ||0||e = 0.

To prove the homogeneity of [|г||е let us remark that for Я Ф 0 we have

n n

INI* = inf UK, yf)II, : J T 1 \ ~

г = 1

» yrfeZf

and (xi} у{)€(Х X Y)e for i = 1 , 2 , Writing xt = xJA, yi = y{, we have (xif ^ ) e ( I x Y)g, and so

71 П

IN Ie .= in f { | A | - ^ | | K ,

г= 1 г=1

and К , yi) e { X x Y ) e for i = 1 ,2 , .. . , nj = |A |-|K , since ||(Я^, &)||e = |Я| ||K , ^)[|e.

Finally, the triangle condition is obtained as follows, where the points under investigation {xt , yf), (х'{, у\), (x", у") are supposed always to belong to ( Z x Y)e :

П П

H « i + * a | | e = ^ f { ^ | | K , 2/г) ||е:

г=1 г = 1

< in f

Ц К ,

+

Ц К ',

г —1 г = 1

P Я

JS1 К , yfcZu £ (XÏ, yï ) е02}.

г=1 г=1

Now, we pass first to the infimum with respect to all sums я

y'i)ez2, obtaining

i = i

1 У _

||г||е < in f \S ?/г)lie~I- IItalie * О^г?

e^lj

г=1 i'=l

P P

=

inf llK>

+ IM e = IHille-HMe-

1.8. Let us now suppose that q is a bipseudomodular in X x Y T where X is a real linear space, and Y is a real Banach space. Let us writo

||®||° = sup ||(a?, y)\\e, X° = {x : x e X and ||®||° < 00 }.

lll/IKl

1.9. Y° is a linear subspace of X and Ц-Ц0 is a pseudonorm in X й, If. moreover, q is a bimodular in X x Y , then || • ||° is a norm in X°.

P roof. Obviously, 0 eX° and ||0||° = 0. If xx, xzeX°, then ||(®i + + жа>У)11р< 11(®1>У)11е+11(®а>У)11в< M ° + M ° for H 2 / I K I . Hence IK + + ®2||°< ||я?1||°+ 1Иа№- Let a be real, x e X 0. Then \\(ax,y)\\e = |a| ||(®, y)||?

< |a| ||®||° for II 2 /H < 1; hence ||a®||° < |a| ||®||°. Moreover, if a Ф 0, we have

|a| ||®||° = |a| ||a_1(a®)||° < ||a®||°. Thus ||a®||° = |a| ||®||°. How, let q be a bimo­

dular, and let ||®||° = 0. Then ||(®, y)\\Q = 0 for every y e Y, and 1.4. 4*

implies g{x, y) = 0 for all y e Y. Consequently, x = 0.

2. In this Section let E, F be two real Banach spaces, and let Y =

£?{E,F) be the Banach space of bounded linear operators from E to F.

Moreover, let X be defined as follows. We take a measure space (Q, X, //), where y is a measure on a u-algebra E of subsets of an abstract set Q. X is defined as the set of all vector-valued functions defined in Ü with values in E such that the function yx(t) is strongly //-measurable as a function from Q to F for every y e Y. Equality x — 0, where x = x(t), means that yx(t) = 0 almost everywhere for every y e Y separately. Besides X we shall need the space Z of all strongly //-measurable vector-valued functions defined in Q with values in F, with equality of elements defined as equal­

ity almost everywhere.

Let us remark that in case when F — В is the space of real numbers r then Y is the strong dual of E, and the measurability assumption on x e X means weak measurability of the functions x(t). Z is then the space of real­

valued //-measurable functions on Q.

The tensor product X<g> Y will be defined in this Section by means of the formula

П П

(Fi. Уг)®г ~ Уi i I) •

i= ^{1 i= l}

Then the elements of J ® Y are ^-valued //-measurable functions

n

£ y i xi(t). where y ^ Y and x ^ X . Evidently, Y®Y с- Z.

i= 1

2.1. Let

be a convex pseudomodular in Z. Then

q {%, y) = ê W

is a bipseudomodular in X x Y , and ||®||° = sup||//®||~. If, moreover, q is

a modular, then

is a bimodular. I!î/Il<1

Proof. The fact that

is a bipseudomoclular in X

Y is evident.

The relation ||ж||° = sup||y;r||~ follows from the equality \\{®,y)\\Q = \\yx\\~.

HwlKi

Now, let q be a modular, and let

y) = 0 for all ye Y. Then g(yx) = 0, i.e. yx{t) = 0 almost everywhere, but this means that x = 0 in X.

2.2. Let q be a convex modular in Z and let Z~ be the repective modular space with norm ||,*||~. Let g(x, y) = g(yx). Then (X(&Y)e czZ~, and this injection is a bounded operator from (X(g>Y)e to Z~. Moreover, ||г||~ < ||£||e for every ze(X<g>Y)e.

ft

Proof. Let Z€( X®Y) e, i.e. z = J jy ^ ^ t), where д(Ххг, у ^ - > 0 as

i=l

A->0 + for i = l , 2 , . . . , n . Then £ (Л у ^ )-> 0 as Л->0 + for all i.

Thus,

( ^ft

— Я з ) < — V Q i t y i X i ) - > 0

n I n jf-J

as X - > 0 + , i.e. zeZ~. The continuity of the injection of (J® Y )e into Z~

will be proved if we show the inequality ||г||~ < || 2 ||e to be true for ze {X® Y) . Pirst, let us see that if {xi , y f ) e( Xx Y)e; then

II(®<, Уд lle = in f > 0 : Q l ^ - \ < l j = ||y ^ ||~ .

Now, if z e(X(S)Y)e and z(t) = ^jy^x^t), where {x{, y{) e( X x Y)e,

i=l

n n

then we have ||z||~ = ||1 > г жг||е < Е\\У№\\%- Hence

г = 1 г= 1

.11% < inf WViXiWç :z(t ) = 2 y Mi t ) and (ж*, у{)е(Х x Y)e}

г = 1 г = 1

ft тг

= inf { £ ll(®i> Vi)\\e: z (t) = and [х{, у {) е { Х х Y )eJ = 1И1,.

i —1 i= 1

2.3. An element x e X belongs to X° if and only if Мтд(Хух) = 0 uni­

formly in the ball ||y |K l. я~*0+

Proof. It follows from 2.1 that жеХ° if and only if there exists a num­

ber К > 0 such that \\yx\\~ < К uniformly in the ball ||y|| < 1, i.e. g(kyx)

< 1 for a h > 0 uniformly with respect to у in the ball Ц 2 /Ц < 1. Since is convex, this condition is obviously equivalent to limp (Луж) = 0 uni­

formly in ||y|| < 1. л_>0+

2.4. Let (Фх)у — yx for хеХ°, y e Y. Then d>eüf[X°, S£{ Y, Z~)], and the

ПОГт ||^ ||js

?[

°,

2?(Y,Z~)] — 1-

Proof. It follows from 2.3 that if ж е ! 0, then yxeZ~ for every ye Y.

Since the linearity of Ф is obvions, the theorem follows from the equalities

\\Фя\\щТ,г~) = sup||(<&r)y!|~ = sn p H rt- = |N|°.

Q \м<1 Ilî/IKI

3. Now, we shall assume the assumptions and notation of 2, and that E is isomorphic with a part of F : let p be the isomorphism of E into F.

We denote by <p(F) the image of F in F. Let Jc > 0 and К > 0 be such that

^IINL e ^ \\<P x \\ f ^ K\\ x \\ e for XeE.

The following space X~ will be of importance:

X~ is the set of functions zeZ~ such that z(t)ep(E) for almost all t eQ.

3.1. We have \\<px\\~ < JX||a?||0 and <p(X°) с X ~ , where p(X°) is the set of functions of the form <px(t) with xeX°.

This follows from the relations

N1° = s u p | M ~ II2/IK1

p 1

— x\ = —

К \~ в К

IM ê •

3.2. Let the modular q in Z possess the following property:

(*) if z2eZ and H^i(11^ ^ (^)!1г^ almost everywhere in Q, then Q(*i) < q M -

Then the spaces X° and X~ are isomorphic and &||a?||° ^ Ьрх\\д ^ -ЙПИР*

Proof. Let xeX~ and let ||y|| < 1, y e Y. Then

and by (*), we obtain д(Лух) < q(Xk~lpx) -> 0 as A->0 + . Hence yxeZ~„

Moreover, \\yx\\~ < Яг1 ||<yr||£. Thus ||ж||° == sup||yx||? < Х_1||ож||~ < + oo.

ilî/IKi

This together with 3.1 yields 3.2.

3.3. Let the modular ~ q in Z possess property (*) of 3.2, and let it satisfy the following two conditions:

(a)

Ü,

zneX~,

< lim êK ), n-> O O

(b) if \\zn\\~ -> 0, zneX~, then \\zn(t)\\F -^0 in measure y.

Then the space X° is complete.

Completeness of X~ is proved in the classical way; completeness

of X° follows, since X° and X~ are isomorphic.

4. A generalization of the bimodular space considered in Section 2 may be obtained as follows. Let (Ü, 27, /«) and (Qx, Xx, y x) be two measure spaces, and let E, F be two real Banach spaces.

We define Z as the space of vector-valued functions defined on Q x Qx with values in F, strongly measurable with respect to the product measure y X p x, with equality of elements defined as equality j uXyx-al­

most everywhere.

Now, let and Y be the spaces of /q-measurable real-valued func­

tions and of strongly /^-measurable vector-valued functions with values in J?{E, F), respectively, defined on Qx. We denote by X the space of vector-valued functions defined in Q with values in E such that yxeZ for every y e Y. We shall write x = 0, if yx = 0 in Z for every y € Y. Finally, we denote by X x the space of vector-valued strongly /«-measurable func­

tions defined in Ü with values in F.

Let us suppose, a convex pseudomodular q is defined in X x such that

every

e Y, and let q be a convex pseudomodular in

R.

4.1. The formula

< p ( x , У ) = ( Q O Q ) ( y x )

defines a bipseudomodular in X x Y; here, qoq is the superposition of q

and ~ q . Moreover,

INI0 = sup{||(a?, y)\\Q : j|||y(s)||^ ,F)j|ë < !}

is a pseudonorm in the linear space X° = {x : x e X and ||ж|]° < oo}. There holds also ( l® Y ) e c Z~~, and |) 2 ||-0~ < |] 2 ||0 for z e ( X®Y) e. Finally, an element x e X belongs to X° if and only if \im(QOQ)(?<.yx) = 0 uniformly

я^о+

in the ball |j||y (s)||j?(B,F)||ë < !•

This statement is proved analogously to 1.9 and 2.1-2.3.

5. To get an illustrative example of a modular q from Section 2, one may take a convex ^-function and write

q (*) = f 9г(11«(<)11)Ф- Q

Then Z~ is the Orlicz space L*41 (F) of strongly /«-measurable and absolutely 95-integrable functions on Ü with values in F. (X<g>Y)e is the

П

space of functions of the from

=

where for

i = l

i = 1 , 2 , . . . , n, and X° is the space of functions x e X such that

I q ) { h \ \ y x ( t ) \ \ ) d y

< 1 . Q

for all \\y\\ ^ 1 Я|Д(1 âj constant Jc ^ 0 independent of y*

An example to Section 4 is obtained taking

É?0»i) = Q(Vi) = f <Pi(\yi{s)\)dVi-

Q Q1

References

[1] G. K o th e , Topologische lineare Baume, Vol. I, Springer-Verlag 1966.

[2] J. M n sie la k , On some modular spaces connected with strong summability, The Mathem. Student 27 (1959), p. 121-136.

[3] — and W. O rlicz, On modular spaces, Studia Math. 18 (1959), p. 49-65.