Certain classes of polymodular spacesAbstract. The so-called multimodular pseudotopologies and spaces are defined. Lattices of

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria l: PRACE MATEMATYCZNE XXIX (1990)

Ed w a r d Am b r o è k o

(Poznan)

Certain classes of polymodular spaces

Abstract. The so-called multimodular pseudotopologies and spaces are defined. Lattices of these pseudotopologies are considered. A theorem on generating multimodular pseudotopologies by families of filters is given. The Orlicz topology is constructed for our pseudotopologies.

In [3] the so-called polymodular spaces were defined. Here we will consider some special cases of those spaces (multimodular spaces). Our new spaces are such that the Orlicz topology exists for them. Recall a polymodular space can have no Orlicz topology ([1], 5.4.5, 5.4.9; [4], 3.1.4); however, it has always the Orlicz pseudotopology ([4], 3.1.3).

The terminology and the notation of [3] (and [1], [2], [4]) will be kept.

Let X be a linear space over the field К of all real or complex numbers.

1. Polymodular spaces. In [3] a polymodular space was defined as a special case of linear-pseudotopological spaces. Therefore we collect some remarks on these spaces.

Every linear pseudotopology т on X satisfies the following conditions (cf.

[5] , [6] or [7]):

(a) if F (X)э ^ э ^ е т (0), then S '

(0);

(/?) if S ', ^ет(О), then S' глУ

(0);

(1) if S'

Щ, then S' + 3

(0);

( 2 ) i f Xe

K, S'

then X-S^

(3) if S' ет(0), then V-S'

0);

(4) if

X, then F-xei(O);

т(х) =

) + %:= {S'

^ет(О )} for

Here the symbol F(X) denotes the set of all filters in X, V denotes the neighbourhood filter of OeX (К is equipped with the usual topology). Of course, filters of

(

) are said to be convergent to x in the space (X,

).

If a set

satisfies conditions (a), (/?), (l)-(4) and if

т(х):=

+

for

then т is a linear pseudotopology on

(cf. [5],

[6] or [7]).

We denote by LP(X) the set of all linear pseudotopologies on X and by LT(X) the set of all linear topologies on X. We know that if ЗГ

LT{X) and if

t

(

): = { ^

F (I):

+ where 'V is the neighbourhood Filter of OeX with respect to then т

LP(X)% , and we may write: т = ZT. If

LP{X) and if P) T (0) e

(0), then the family f -) т (0) ( = infFW т (0); F (X) is partially ordered by inclusion) is the neighbourhood filter of OeX with respect to a certain linear topology (cf. [5] or [6]).

Define the following partial order in LP (X):

a <

if and only if т(0)

cr(0), a,

LP{X)

(if a,

LT{X), then a ^ z means that cr <= r); let us recall a theorem on the set LP(X) ([2], 5):

Suppose that (rj}jeJ cz LP{X), J ф 0. Then infiJ5 sup

LP(2Q exist (we

j e J j<=J

also use the symbols i n f ^ ^ r 7-, supLP(Ar)Ty) and

j e J j e J

(infT,)(0) = {&

F (X): F => ^ + ... + ^„ for some

j e J

^ , . . . , ^

( J

,.(0)},

j e J

(sup T j) (0) = П т 7 '(°)-

j e J j e J

(So LP(X) is a complete lattice.)

Now recall some facts concerning modular filters ([2]).

For a set A in X we define

A(A):= I- A + I-A , where

I : = { l E K i

and for А с X , s > 0 we define

FS(A)— {xx + fy: x, y e A; a, £

K, |a|s + |jS|s ^ 1} (Г(А):= Г^А)).

If F

F(X) and if s > 0, then the families {A(F): F

F ] , {TS(F): F

F ] are filter-bases in X; therefore we can define the following filters generated in X by these filter-bases:

A

F

F}~\ ( = I - F + I - F ) , C{FS(F): F

3?\\

(Г (#-):= Г , (#■)).

Let a number у > 0 be given. We say that fF

F {X) is a (generalized)

modular filter of character у in the space X if it satisfies the following

conditions:

y& c= A ),

& <= Vx for

X.

We denote by FM y(X) the set of all modular filters of character y in X.

A filter ^

F (X) is said to be s-modular (s > 0) if 3F c= Г3 {3F) and f c be for

X. We denote by FSM (X) the set of all s-modular filters in X.

A filter-base in X is called a locally s-convex modular base (s > 0) in the space X if every set of & is absolutely s-convex and if for any B

^ ,

X there exists a number а ф 0 such that ax

B. If is a locally s-convex modular base in X, then the filter [J>] (generated in X by Щ is called a locally s-convex modular filter. The set of all locally s-convex modular filters in X is denoted by F M ^ i X ) .

We say that a linear pseudotopology i o n I satisfies condition (M) with a filter & (ex(0)) if ([1])

x(0) = {&

F(X): X3* for a certain Х

К (X Ф 0)};

we write here x — x^.

Let numbers y, s > 0 be given. A linear pseudotopology x on X is said to be modular of character y, s-modular, modular and locally s-convex if ([2]) there exists a filter & EFMy(X), &

F.

M(X), &

F M

_

(X), respectively such that x = ijsr. The symbols PMy(X), PSM{X), PMS_C (X) denote the set of all modular pseudotopologies of character у on X, the set of all s-modular pseudotopologies on X and the set of all locally s-convex modular pseudo­

topologies on X, respectively ([2]). Recall that if y e (0 ,2), then PMy(X) = LT(X).

Take a number у > 0. We say that a linear pseudotopology x on X is polymodular of character у ([3]) if there exists a nonempty set G cz PMy(X) such that x = infLPm G. The set of all such pseudotopologies on X is denoted by PpMy{X). If x = infLPW G, where G c: LT(X), G Ф 0, then x is called a poly-LT-pseudotopology.

Recall also ([3]):

PpLT(X):= P pM ^X ),

PpsM (X):= {inf„m G: G c PSM(X), G # 0 } (the set of all poly-s-modular pseudotopologies on X; s > 0),

PpM,.JX)-.= {inf„w |G: G c PMS. J X ) , G Ф 0}

(the set of all locally s-convex polymodular pseudotopologies on X; s > 0), PpMM(X):= {inf„m G: G c (J PMy(X), G Ф }

у > 0

(the set of all polymodular pseudotopologies of a floating character on X).

2. Multimodular spaces

2.1. Let y be a positive number.

We say that a linear pseudotopology т on X (and the space (X,

)) is multimodular of character y if there exists a nonempty set G a PMy (X) such that

= supLP(X) G.

Observe that if G = 0 , then supLP(X) G = т{Х} (the anti-discrete topology on X). Of course, the topology

{X} is multimodular of character y; we have T{X} = supLP(X) G, where G := {тда}, т{Х]

РМ

(Х). If G cz PMy(X), G ф 0, and if т = supLP(X) G, then we may say that the multimodular pseudotopology т is generated by the family G of modular pseudotopologies.

2.1.1. Recall (e.g., [2], 5.1.1) that if G c LT(X), then supLP(X)GeLT(X). In other words, a multimodular pseudotopology generated by linear topologies is a linear topology. Note also that the supremum of a family of modular pseudotopologies need not be a modular pseudotopology ([4], 1.4).

2.1.2. Let us denote by PmMy(X) (y > 0) the set of all linear pseudo­

topologies on X which are multimodular of character y.

2.1.3. The set PmMy(X) (y > 0) is a complete lattice.

P ro o f. Consider a set H cz PmMy(X), H ф 0. Let a pseudotopology т е Я be given. Then we have т = supiPW GT, where Gx is a certain nonvoid subset of PM y{X). Of course, we get

t

(0) = П *(0).

o e G T

Therefore

(

P

P(

}

(0) = П T(°) ⁼ П ( П ff(°)) = П ff(°) with ^{G: =} U ^Gf

teH reH oeG T oeG teH

Obviously, G c PM y (X), G Ф 0; hence

supLp(X) H e PmMy (X), supLPm H = supPmMyW H.

Of course, the anti-discrete topology x{X} belongs to PmMy(X) and

= m^pmMv(X) PmMy (X) = supPmAfy(X) 0.

Now we see that

iniPmMv(X)H = supLP(X) {<7 ePmM y(X): а ^ т for all теЯ }.

2.1.4. Observe that if Я c PmMy(X), H ф 0, у > 0, and if for т е Я we have

supLP(X)Gx, where Gx

P M y(X), Gx Ф 0, then

supLPWЯ = supLP(X)G with G : = { J G X.

zeH

2.1.5. If a pseudotopology is multimodular of character у > 0, then this pseudotopology is polymodular of character y (cf. [3], 1.1.2, 1.1.4).

Therefore

PM y(X) cz PmMy{X) c PpMy(X).

Obviously, the equality PmMy (X) = PpMy (X) need not be satisfied; e.g., the poly-LT-pseudotopology of [4], 1.5, is polymodular (of each character y > 0) but it is multimodular of no character ye(0, 2). For y e (0,2) the equality PmMy(X) = LT(X) holds (2.1.1).

Note that if 0 < yx < y2, then

PmMyi(X)czPmM y2(X) (cf. [2], 1.3.4).

2.2. Let a number s > 0 be given. Consider the set of all multi-s-modular pseudotopologies on X

PmsM (X ):= {suP u w G: G <= P,M (X), G Ф 0},

and the set of all locally s-convex multimodular pseudotopologies on X PmMs_ç (X) : = {supLF(„ C : G <= PM „JX ), G Ф 0 ).

Obviously, the topology x{X} = supLP(X) 0 is a locally s-convex multimodular pseudotopology for any s > 0, i.e.,

T{X)EPmMs_c {X), and hence

PmsM (X).

2.2.1. The sets Pms M (X), PmMs_c (X) (s > 0) are complete lattices (cf.

2.1.3).

2.2.2. Note that for s > 0 we have

Ps M (X) c Pms M (X) c= PmM2i/s (X) n Pps M (X) (cf. [2], 1.3.7) and

PAf s-c. (X) c PmMs_c (X) c Pms M (X) n PpMs.c. (X) (cf. [2], 1.3.11). Moreover, observe that if 0 < st < s2, then

PmS2 M (X) c PmSl M (X), PmMS2.Ci (X) <= PmMSi.c. (X) (cf. [2], 1.3.6, 1.3.12).

2.2.3. If s > 1, then

(cf. Г2]. 1.3.14. (2)).

PmsM(X) = LT(X)

2 — Roczniki PTM — Prace Matematyczne XXIX

2.3. Note that a polymodular pseudotopology of character у > 0 on X can be represented as the infimum (constructed in the complete lattice LP (X)) of a certain set of modular pseudotopologies of character y; a multi- modular pseudotopology of character у > 0 can be represented as the supremum and as the infimum of appropriate sets of modular pseudo­

topologies of character y. Analogous remarks are true for multi-s-modular pseudotopologies and for locally s-convex multimodular pseudotopologies (s > 0).

3. Generating multimodular pseudotopologies by families of filters. Con­

sider a mapping

со: X

where (JX6A- со (x) Ф 0, and let

m := IJ (co(x)-x),

x e X

where

co(x) — x: = {Jf — x: ^ eco(x)}, x e X . Moreover, take numbers y, s > 0.

There exists a unique pseudotopology xeP m M y{X) (

ePmsM(X), x e PmMs_c (X)) such'that d i e x (0), i.e., co(x) a x(x) for x e X , and that a ^ x for every a e PmMy (X) (<x e Pms M (X), a e PmMs_c (X)) with a (0) со.

P r o o f (for PmMy(X); cf. [4], 2.1). We have x = supLP(Ar) {<7

PmMy (X): <r(0) => œ}.

Observe that the set I : = (cr e PmMy (X): a (0) о d>) is nonempty because the anti-discrete topology x{X} belongs to I .

4. The Orlicz topology. We say that a linear pseudotopology x on X is equable ([6], [7]) if for each ^ e x ( 0 ) there is a filter ^

(O) such that УУ c Recall (cf. [7], 5.8) that for a pseudotopology

LP(X) there exists a unique equable linear pseudotopology

# ^ x ( x * e L P ( X )) such that for every equable linear pseudotopology a ^ x the condition x* ^ a holds; we have

t

# (0) = {#"

F(X): there exists a filter ^ет(О) with V<S a J 4 5"}.

4.1. Let x be a multimodular pseudotopology of character у > 0 on the space X. Then the pseudotopology

# satisfies the following conditions:

1° x* ^ x;

2°

# is a linear topology;

3° x* ^ о for every linear topology a ^ x (<r

LT(X)).

P ro o f. Suppose

т = supLPm гj, where Xj e PMy (X)

je J

for j e J ф 0. We know that

(supLPm(Tj)* = swpLP(X)o f ,

j e J j e J

i.e.,

(supLPW <7j)# (0) = p o f (0)

je J je J

for O j€L P (X ),jeJ (cf. [7], 5.12). Obviously, property 1° is a general property of the operation #. Since Xj (je J) are modular pseudotopologies, x f are linear topologies ([1], 5.4.5); therefore

T # = supLPW T f jeJ

is a linear topology (2.1.1). Hence property 2° is true. Now consider a linear topology о e LT{X) and suppose that о ^ x. Of course, the topology о is equable; therefore

# ^ o. Property 3° is proved.

4.1.1. The topology z v := z* considered above will be called the Orlicz topology for the multimodular pseudotopology z.

4.2. In [4] so-called Orlicz pseudotopologies were constructed. Recall the adequate theorem ([4], 3.1.1):

Suppose that z e PpM(y)(X). Then the pseudotopology

# z) satisfies the following conditions:

1° z*eP pLT{X );

2°

# < (j for every о ePpLT(X), о ^ z.

The pseudotopology z* constructed in this theorem is also denoted by zp\

Now we see that if a linear pseudotopology x is multimodular, then its Orlicz pseudotopology is a linear topology (here we have

# = zpv = i v).

Let us give the following cofollaries:

4.2.1. If a poly-LT-pseudotopology z is not a linear topology, then z is not a multimodular pseudotopology (because z* =

i.e.,

xeP pLT(X)\L T(X ) => хф (J PmMy(X).

y > 0

Observe that from 2.1.1 we have the implication

xeP pLT(X)\LT(X ) => хф U PmMy{X) (= LT(X)).

y e ( 0 , 2 )

4.2.2. A multimodular pseudotopology (of a certain character) is a linear

topology if and only if it is equable.

5. Examples

5.1. Let X be the linear space of all bounded sequences of real numbers.

Take the sets

F„:= {(* 1 , ...)eX: |x j, |xB+1|, ... < 1}

and the filters

where ne N (the set of all positive integers). Define V = V „ > n e N ([4], 1.3) and let

t

:= m{LP(X) тп.

neN

Of course, & n + l c therefore and oo

T( 0 ) = u *.( 0 ).

n= 1

We construct the Orlicz pseudotopology i pv. One has

tpv

=

= mfLP(X)x*,

(0) = (J

(0).

neN n = 1

Obviously, г* = TFJE-n, n e N ([1], 5.4.5). We have

V P , = [{{(x„ ...)eX: |x„|, |x„+1| , ... < e}: e > 0}].

Hence

* is the linear topology generated on X by the seminorm IMI„:= sup{|x„|, |*„+1| , ...}, x = {xt , ...)e X .

Since VlFH + ! c V ^ n, we get

„#+

^

* . Moreover,

„#+

ф т„# . Now it is easy to see that r # does not satisfy condition (M). Therefore т* фЬТ(Х), and hence (see 4.1) хф[]у>0 PmMy{X) (i.e., the pseudotopology т is not multimodular).

Consequently, we may write

t

e PpMt_c {X)\ (J PmMy(X).

y > 0

Recall that in [4] we obtained

г е Р р М ^ 'Ю Х Р р Ь Ц Х ) .

5.2. Next, let X be the linear space of all sequences of real numbers. Define (see [4], 1.4)

F„:= {(xj, ...)eX: IxJ, ..., |x j

1}, &„: = [ F J ,

n e N .

Moreover, let

т: — supLP(X)T

neN

The pseudotopology x is not modular but, of course, it is multimodular. Let us construct the Orlicz topology for the pseudotopology x. Denote

:= V & n (= [{{(*!, ...)eX: | x j , ..., \x„\ < e}: s > 0}]).

We have

T * = V„v » I v =

# = supLP(X) т * .

neN

Obviously,

tv

(0) = {&eF(X): <S => ^ } , where

# ":= s u p ^ J 2^ = [{{(%!, ...)eX: |x j, ..., |x„| < e}: e> 0, neN }].

neN

It is obvious that x v ( = x^) is the linear topology generated on X by the F-norm

*11

=

{хг, ...)e X .

n = 1

References

[1] E. A m b r o z k o , Pseudotopologies for modular spaces, Comment. Math. 25 (1985), 189-200.

[2] —, On modular filters, ibidem 29 (1990), 1-21.

[3] —, On polymodular spaces, ibidem, 29 (1989), 23-31.

[4] —, Examples and constructions o f polymodular pseudotopologies, 29 (1990), 117-127.

[5] H. R. F is c h e r , Limesraume, Math. Ann. 137 (1959), 269-303.

[6] A. F r o lic h e r , W. B u c h e r , Calculus in Vector Spaces without Norm, Moscow 1970 (in Russian, translated from English).

[7] W. G â h le r , Grundstrukturen der Analysis II, Berlin 1978.