I kl < + oo.

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

Ed w a r d Am b r o z k o (Poznan)

Examples and constructions of polymodular pseudotopologies

Abstract. In [3], the notion of polymodular space was introduced. Here we give some examples to illustrate our theory. Moreover, we prove a theorem on generating polymodular pseudotopologies by families of filters and define so-called Orlicz and Wiweger pseudotopologies.

In the paper we use the terminology and the notation of [1], [2] and [3].

Let the letter X (as in [2] and [3]) denote a linear space over the field К of real or complex numbers.

1. Examples.

1.1. Let X be the linear space of all sequences (x1} x2,...), where 00

•xl5 x 2, . . . e R (the set of all real numbers) and following filters in X:

I kl < + oo.

?,'■ = [{{(*!,•■•)£*: I № < e } : s > 0 } ] , n eN : = {1, 2 ,...} .

k - n

For n e N we have R F n = + and !Fn a Vx, x e X (recall that /: = {ÀeK: \À\ ^ 1} and V is the neighbourhood filter of 0 in К with the usual topology). Observe that !Fn is the neighbourhood filter of 0 in the seminormed space {X, || y , where

00

||(xi,...)||„:= X Ы , (x u ...) e X (|| ||! is a norm).

k = n

Let

V = V „ (see [2], 1.3), r := mfLP(X)xn (see [2], 5).

neN

{(xl t ...)eX : X \x k\ < £} Œ { ( * ! , . Z i^l < £}

k=n k=n+l

Since

118 E. A m b ro z k o

for £ > 0, we have ^ n + x cz 3Fn, and hence t„ + 1 ^ t„ (neN). Therefore, I = indlimT„,

ne N

t(0) = (J t„(0) = {& eF(X): 3F => for a certain neN }

ne N

([3], 1.2.1; F(X) denotes the set of all filters in X). Obviously, т is a poly- LT-pseudotopology ([3], 1.1.1). Assume т satisfies condition (M) ([1]) with a filter &. Then there exists a number n0e N such that & no cz 3F (recall # r„0 is an LT-filter). Thus for every ^ет(О) one gets a number 2 ^ 0 for which

^ 3 X& 2,#'яо. Therefore ^ „ 0 c ^ for each ^ex(O). In particular, we obtain the inclusion J*„0 с &гио + 1. Define e„:= (x1? ...), where x* = 0 for к ф n and xn = 1 (n, keN). Since

{(*!, ...)e X : ^ N < 1 ) 6 ^ ^ # ^ ! ,

к - n o

there exists an e > 0 such that

00 00

{ (* !,...)eX: ]Г \хк\ < б} C= {(x1? ...)eX : J |x j < ^

к = n o + 1 к = no

But this inclusion is impossible because, e.g.,

00 00

e„o6 {(*i» •••)£*: Z \xk\< e } \ { ( x 1, ...) e X : £ |xk| < 1}.

к = n o + 1 k = no

Therefore the pseudotopology т does not satisfy condition (M), and

tePpLT(X)\LT{X), i.e., т is a poly-LT-pseudotopology and is not a linear topology.

1.2. We continue discussing the pseudotopology т of 1.1. Consider the filters

W < 1, X \xk\ < s } : s > 0}], neiV.

fc = n + l

One sees that 1Уп = &„, = 2&n and <S n <= Vx for x e X . Therefore the filter (S n is modular of character 2 (moreover, is a locally convex modular filter). We show that is not an LT-filter. Let

00

A : = { ( Xl, . . . ) e X : |x „K 1, £ W < •}■

fc = H+ 1

Of course, A e & n. Observe that 2ene B \ A for each Ве2У п. Therefore,

^ „ ф 2^„, and hence is not an LT-filter. We shall show that J F+ 1 cr (3 n cz 3F n for n e N . Let a set F e tF n + x be given. Then there exists a number £ > 0 such that

F

3

X k l < £}-

k = n+ 1

Moreover,

00 00

{(xl5 . . . ) e X : X W < e } => { ( *1, l*J < 1, X k l < e} e ^ „ .

к = л+ 1 fc = n+ 1

Therefore Now let G e^„. Then

00

G => { к , |xB| < 1, J] |xk|< e } fc = n + 1

with some ee(0, 1). Furthermore,

{ k , ...)eX : |x„| < 1, £ W < e} => {(xl5 X k l <

k = n + 1 к = n

so G e ^ n. Hence one may write

^и+1 C

c ^ wc ^ n, + neiV.

Let

<V = <*:= inW ) an (= ind lim <rn).

neiV neW

We have

<7(0)= O „ ( 0 ) = { ^ e F ( X ) : neN

there exist an n e N and a l e R such that $F =э Я^п}.

We show that а — x. Since c we get cr„ ^ t„ (neN). So а < <т,( ^ т„, and consequently g ^ x. Moreover, <7Я ^ т л+1 ^T . This implies g ^ t. Thus we obtain the equality <x = x. Hence one has

t = infLP{X)GnePpLT(X), апфLT(X) for n e N .

neN

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

Consider the sets

F„:= { k , ...)e X : \xn\, k + il* < 1}

and the filters

^ n - [ F n l ,

where ne N . Observe that Fn c Fn + l ; therefore we have ^ n + 1 <= n (neN).

Moreover, r ( F n) = Fn ([5], [1] or [2]). Fix x = (x1, . . . ) e l Since x is

120 E. A m b roz k o

bounded, there exists an M > 0 such that IxJ, |x2| , ... ^ M, and hence (1/M) x e F n. Of course, the set Fn is balanced; therefore the relation (1/M) x e F n gives the inclusion [ F J c Vx. Noting the above facts we see that & n is a locally convex modular filter (and therefore e FM 2 (X); see [2]). Observe that F J 2 & n. Therefore & n$FLT(X) (= FM 7(X), where ye(0, 2); see [2]).

Let us define

T n : = z : = L P ( X ) T n ■ neN

Since J 57n + 1 c= & ni we have

T„+ 1 <T„, T ( 0 ) = U '.( 0 ) ([3], 1.2.1).

neN

Assume that the pseudotopology т satisfies condition (M). Then ^ no+1 => X^ n0 for certain n0e N and XeR. However, one sees that X ^ n ф # ”n + 1 for n e N and XeR. Indeed, (|2| + 1 )ene F n + l \XFn, and hence XFn j>Fn+1 {neN, XeR).

Therefore the pseudotopology т does not satisfy condition (M). Moreover, the property X3Fn ф +1 (neN , XeR) implies that т„ ^ тn + i {neN). We will show that T is not a poly-LT-pseudotopology. Assume that т ePpLT{X). Then for every n e N there exist a filter УneFLT{X) and a number kne N , kn >n, for which 3Fn Уn From the condition => 3F kn one has

= V(S n 3 F ^ kn. Therefore FJ%n c= Since then < — h i ) Fkne F J5^ c= ^ n, we get < — j, 2) Fkn => F„. One has obtained a contradiction because, e.g., ekneF„ and ekJ <—i , i> Fkn = {(x1? .. .)eX: | x j , |xkn + 1|, ... < i}. Hence z e P p M l4tm(X)\PpLT(X) (see [3]).

1.4. Now let X be the linear space of all sequences of real numbers.

Consider the sets

Fn:= {{xl ,...) e X : |x j |,..., |xB| ^ 1}

and the filters

* п - - = 1 Ы where neN .

We observe that Г (Fn) = F„ for neN . Fix a number n e N and an element x = (x1, . . . ) e l Let M := m ax(l, |jcx|, ..., |xj). Then

and therefore

3F n c Vx.

We now see that 3Fn (neN) is a locally convex modular filter. is not an LT-filter. Define

T : — SU P L P ( X ) Tn- neN

X : = T

Since F„+ 1 c=F„, one has ^ „ с # л+1 and + We shall show that

t„ + i Ф x„. Assume тп + 1 ^ т „ . This means that ф ^ „ + 1 c= for a certain number £ ф 0. But this inclusion is impossible because £F„+i4>F„ (e.g., 2^en+1e F n\ ^ F n + 1). Let

s f : = {supf m a„SFn\ ctne R , a„ Ф 0, neN }.

neN

Then ([2], 5).

t(0) = {3F e F (X): ^ <= 3F for a certain

Obviously, we have sé c= т(0) n F M l,c (X) (see [2]). Consider arbitrary se­

quences (a„), (j8„), ane R , fineR, and suppose that |a„| ^ |jS„| for neN . Then Pn0 ^«o Œ an0 & no c SUPfix) for each n0 e N;

neN

therefore

supF ( X ) P n ^ n c supFWa„^{J \ .}

neN neN

Put the question: Does the pseudotopology т satisfy condition (М)?

Assume that the answer is affirmative. Then there exists a filter Ж e x (0) such that for every У e x (0) the inclusion а Ж <= ^ holds with a certain number а ф 0. Moreover, one can choose a filter 2Fe s / , с j f . Hence for each

e x (0) we have a3F c= where a is some number different from zero.

Therefore, in particular, for every ^ e s é the inclusion c ^ holds with an appropriate number а Ф 0. However, we will show that for every & e sé there exists a filter <êesé, с such that а # - Ф ^ for a Ф 0 (of course, if

^ e s é , then 0 # - = [0] ф У). Take a filter

^ = supF{X)<xn^Fnes#, ane R , ccn Ф 0 for n e N .

neN

Let : = la j + ... + |a„|, where neN . Then we have |ocn| ^ iin (n e N) and 0 < fi1 < ц2 < ... Therefore

& Ж :=supF(X)ii n& n.

neN

We prove that there exists a filter ^ e s é such that <хЖ Ф ^ for a # 0 (obviously, if a Ж ф then a # - ф ^). Таке Xn: = пцп (neN). Of course, we have 0 < Xx < X2 < ... Let

^ : = s u p FWA„JV

neN

Since 0 < ^ Xn for n e N, the inclusion ^ cz Ж holds, and therefore ^ c . We want to show that а Ж ф ^ for a e R, a Ф 0, i.e., that for each a e R, a Ф 0, there is a set A e a J f such that for every Be У an element belonging to B \ A

122 E. A m b ro z k o

can be found. Let a number a e R, a Ф 0, be given. Let A : = a# , where H: = pnF„ and n > |<x|. Obviously,

A e a J f, |a/i„| < Щ„ — Я„, A = {(xt , ...)e X : K l , ..., \xn\ ^ \ctpn\}.

Let В be an arbitrary set of the filter Then there exists a positive integer k > n such that В з D : = Ât Ft n ... n Xk Fk. Observe that if D Ф A, then В ф A (if В <= A, then, by virtue of the inclusion D с B, we have D c= A). Let y = ( j j , . . . ) £ X Obviously, y e D if and only if

Lvil <

W < Я2, \y2\ ^ Л2,

I.Vll ^ ^fc, 1.^21 ^ ^k> • • * » IXfcl ^ ^к>

i.e., if and only if lyj < Я15 |y2| < Л2, ...» |yk| < A*. Let z = (zx, ...):= A„é?„. Of course, zeZ). Recall that A = {(х1? | x j , ..., |x„| ^ |a/i„|}. Moreover, z„ = An > \otpn\; therefore z$A , and consequently D ф A. We now see that the polymodular pseudotopology x (it is locally convex) does not satisfy condition (M). Therefore

T

U

у > 0 Furthermore, т has the form

T = supLPmT„, xne P M 1_c (X) (see [2]).

neN

Recall that if H cz LT(X), then supLP{X)H e LT(X).

Let У be a linear space and let o t , a2 be linear pseudotopologies on У. If

ctj ^ g2 and (jj Ф g2, then we write g x < g2 .

Continue our investigations. We have shown that for every filter ЗР es#

there exists а У estf, У c: such that a ф ^ for a Ф 0. Therefore for any 3F e sé there is a filter ^ e s é , c= , with % < i> . Observe that

> x = supLP(X)t„ for all estf.

neN

We will prove that for every pseudotopology q eLP(X) satisfying condition (M) the following implication holds:

if ^ q for n e N , then there exists a pseudotopology

(7бРМ1 ч (X) such that xt , x2, ... ^ q and g<q. Let a peudotopology g e L P (X ) satisfy (M) with a filter ê and let

g ^ x 1, x 2, . . . Then with certain numbers

al , a2, ... Ф 0, and hence

=> jF := s u p F(X)a„JF„.

• > neN

Obviously, 3F e s / c F M t_c (X) and Q = xs '^Xp. Moreover, we can find a filter У e s /, a 3F, such that % < ₁> . Let cr: = We now have

Q — Xg ^ T^ ^ — (7.

Furthermore, т1? т2, ... ^ о. The filters & „ (n e N ) belong to FMS.C (X) for every se(0, 1) (and to FM y(X) for у ^ 2).

Our example shows that the families PMS_C(X) (se(0, 1», PSM (X ) (se(0, 1» and PM y{X) (y ^ 2) are not complete lattices (however, they are lattices; see [2]).

1.5. We give another example of a poly-LT-space.

Denote by LP the linear space Lp« 0, 1)) (1 ^ p < + oo) and let on LP the following norm be given:

W fW p-^lW m V dty1”, f e U . о

Here we say that = f 2 {fx, f 2 e Lp) if and only if (t) = f 2 (t) for almost all te< 0, 1>. Suppose that 1 ^ p' < p" < + oo. Of course, LP” a LP. So one can consider on LP” the convergence generated by the norm || ||p,. Let x’ be the (linear) topology generated on LP" by the norm || ||p, and let x" be the topology generated on LP” by || ||p». Then x' < x" (i.e., т"(0) с t'(0) and т"(0) Ф t'(0));

x' ^ x" because \\f\\p< ^ ||/ ||p„ for feLP ”. Let us give an example of a sequence which goes to zero in {Lp”, x') and does not converge to zero in {LP”, x") ([7], p.

22^):

n 1/p/ l n ( n +¹⁾ for 0 ^ t ^ ^1/и, 0 for l/n < t ^ 1;

Now consider a decreasing sequence (p1, p 2,...), where pl , p 2, . . . e e(l, + oo) and let X : = LPl. Moreover, let т1? ^x2 , ... be the topologies generated on X by the norms || ||pi, || \\P2, ..., respectively. Then we have xl > x2 > ...

Consider the pseudotopology

x : = infLPW xn.

neN

Obviously,

x = indlim xn ([3], 1.2.1), x(0) = [j x„(0).

neN neN

Suppose that the pseudotopology x satisfies condition (M) with a filter &. Then there exists a positive integer n0 such that e xno (0), and hence x (0) cz xno (0).

But this is impossible; therefore x does not satisfy condition ДМ).

124 E. A m b ro z k o

1.6. Of course, polymodular pseudotopologies are balanced. However, there exist balanced linear pseudotopologies which are not polymodular. For example, supLP(X)LP(X), where X is an infinite-dimensional linear space, is balanced and is not polymodular (here (supLP(X) LP (X)) (0) has no modular filter).

2. Generating polymodular pseudotopologies. Consider a mapping

со: X - 2f(*>,

where \J xeXco(x) ^ 0, and let

d>:= (J (ш(х)-х),

x e X

where

co(x) — x: = I # - —x: ^eco(x)} for x e X .

Furthermore, suppose that numbers s > 0 and у > 0 are given.

2.1. There exists a unique pseudotopology т ePpM y{X) ( i ePpsM(X),

te PpMs_c (X)) such that m с т (0), i.e., со(x) c= т (x) for x e X , and that a ^ т for every crePpMy(X) (<rePpsM(X), c e P p M s_c (X)) satisfying the condition (D C O-(0).

P r o o f (for PpMy(X)). Consider the set I of all pseudotopologies стеРрМу(Х) with <r(0) => ю. Observe that the anti-discrete topology on X belongs to I; therefore I ф 0. Let z : = supLP(X) I . Obviously, z e PpMy (X).

Since t(0) = p)ffeI <r(0) and d>c;o-(0) for each pseudotopology aeX, œ is a subset of т (0). Moreover, ^g ^ z for any ^ge l .

2.2. Let us mention the following theorem:

There is a unique linear topology z on X such that œ a t(0) and g < z for every g gLT{X) satisfying the condition <oc:g{0). Moreover,

z = supL T ( X ) {ge LT{X): cb a <r(0)}

(= supLP(X) {o e LT{X): <b с <r(0)}),

t(0) = p) (<t(0): oeLT(X), <b c cr(0)}.

3. Orlicz and Wiweger poly-LT-pseudotopologies.

3.1. Investigate the pseudotopology z* ([4], [1]) fora ze P p M {y)(X) ([3]).

3.1.1. Suppose that zeP p M (y)(X). Then the pseudotopology z* satisfies the following conditions:

1° z*ePpLT(X),

2° t# ^ g for every GePpLT(X), g ^ z; of course, z* ^ z.

P ro o f. 1°: Let a filter ^ ex* (0) be given. Then => for a certain e x (0). Moreover, there exists a filter Ж е х (0 ) n [ J y>0FMy(X) such that

^ Ж. Therefore we have Ж ^ К Ж Furthermore, F J f ex* (0) n FLT(X).

Hence x* ePpLT(X).

2°: Let aeP pLT (X ) and suppose that x ^ ^g. Obviously, the pseudo­

topology g is equable; therefore x* ^ a ([4], [1]).

3.1.2. Let x ePpM {y)(X). Then xeP pLT(X) if and only if t# = x.

3.1.3. The pseudotopology xps/:=x* of 3.1.1 will be called the Orlicz poly-LT-pseudotopology (shortly: the Orlicz pseudotopology) for the pseudo­

topology X (cf. [6], [1]).

3.1.4. Consider the poly-LT-pseudotopology x and the topologies x„ of 1.5.

Moreover, suppose that ^g ^ t, where GeLT{X). Then ^g ^ xn for a certain positive integer n. However, we have xn > i n + 1 > x. Therefore ^g > xn+l > x. Of course, xneLT(X). We now see that for x there exists no linear topology x v eLT(X) which would satisfy the conjunction of the following conditions:

1 ° t v ^ T,

2° tv ^ ^g for any linear topology ^g ^ x (G eLT(X)).

3.2. Construct another poly-LT-pseudotopology for a polymodular pseudotopology. First recall that if a linear pseudotopology x satisfies condition (M) with a filter then # ' л := flt(^ ) ([2], 4.1.6) is the neighbour­

hood filter of zero for the Wiweger topology т л ([1]).

3.2.1. Suppose that xeP pM {y)(X) and let xрл (x):= {Ж eF(X): Ж => У А for a certain filter

^ et(0 )n (J FMy{X)}+x, x e X .

y > 0

Then x ^ хрл ePpLT(X) and xpA ^ ^g for each pseudotopology ^ge PpLT(X) satisfying the condition ^g ^ x.

P ro o f. We show that трл is a linear pseudotopology on X (see [1], 2.1 and 2.2). Conditions (a), (4) of [1] are obvious. Let us verify (/?) and (1). Let filters Ж ^ 2е т рл(0) be given. Then 2Fk з ^ кл for certain

^ kG i(0)n [Jy>0FM y(X), к = 1, 2. Since Ук z> # 4 + # 2, к = 1, 2, we have П ^ ( ^ + ^ 2)л.

Thus

+ 0 2а = > ( ^i + ^ 2 )a+ ( ^i+ ^ 2 )a = ( ^ 1 + ^ 2) л ,

and hence

^ 1 + ^ 2 =>(^1 + ^ 2) \

126 E. A m b r o z k o

Of course,

so

0 , + 0 2ет(О)п U FM,(X),

y > 0

J Jr1+ # '2£TpA(0)

(condition (1) is satisfied). On the other hand, for the same filters & 2> <&l and we obtain

Hence

^ 1 n ^ 2 =>(^1+ ^ 2)A.

This means that n г е х рА (0) (the family трл (0) satisfies condition (/?)).

Now let ^ e F ( X ) and let <F => ^ A, where ^ e i ( 0 ) n 1J FMy(X). Moreover,

у > 0

take a number ЯеК. If Я = 0, then Я#- = [0 ] = э ^ А. For Я # 0 we have Я#" э Я ^ л = (3 A. One now sees that трл (0) satisfies (2) of [1], 2.1. For the same filters ^ we obtain VЖ V(S A ~ ^ A (condition (3) is satisfied).

Observe also that тpA ePpLT(X) and трл < т (because c= ^ for any

y > 0

Now consider a pseudotopology aePpLT(X), suppose that g ^ т and take a filter ^ е т рл (0). Then J* =z> ^ л for a certain ^ e т (0)n (J FM y(X).

у > 0

Since t (0) cz cr (0), the filter ^ belongs to <7 (0). Choose a filter Ж eF L T (X ) n <7(0) for which # => Ж Then ^ з ^ л э ^ л = еа(0), and hence ^6(7(0). We have shown that трл (0) c= cr(0), i.e., а ^ трл.

3.2.2. Suppose that z e P p M {y){X). Then т ePpLT(X) if and only if трл = т.

3.2.3. If x eP pM (y){X), then we say that трл is the Wiweger poly- LT-pseudotopology (shortly: the Wiweger pseudotopology) for т (cf. [6]

and [1]).

3.2.4. Observe that the Wiweger pseudotopology трл (for zeP p M {y)(X)) can be regarded as the poly-LT-pseudotopology generated by t(0) (see 2.1).

3.2.5. Obviously, for %eP pM (y)(X) we can also construct the topology z A of [1], 5.4.8; of course, the equality zpA = т л need not be satisfied.

3.2.6. Suppose that x eP pM (y){X). Then we have:

Г z A ^ трл ^ t ^ xpv,

2° x ePpLT(X) if and only if трл ^ rpv, 3° xeL T(X ) if and only if z A ^ zp\

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 (1989), 1-21.

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

[4] A. F r ô lic h e r , W. B u ch er, Calculus in Vector Spaces without Norm, Moscow 1970 (in Russian, translated from English).

[5] R. L e s n ie w ic z , On generalized modular spaces. I, Comment. Math. 18 (1975), 223-242.

[6] —, W. O r lie z , A note on modular spaces. XIV, Bull. Acad. Polon. Sci., Sér. Sci. Math.

Astronom. Phys. 22 (1974), 915-923.

[7] R. Si к or sk i, Funkcje rzeczywiste, vol. II, Warszawa 1959.