ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXV (1985)
Ed w a rd Am b r o z k o (Poznan)
Pseudotopologies for modular spaces
Abstract. For generalized modular spaces ([6]) a linear pseudotopology ([1], [2], [4]) is defined. A class of linear-pseudotopological spaces satisfying a certain condition (called condition (M)) and including generalized modular spaces is introduced. Balanced linear-pseudotopological spaces satisfying condition (M) are considered. Criteria for linear-pseudotopological spaces satisfying (M) to be linear-topological are given. So-called Orlicz and Wiweger topologies are constructed for linear-pseudotopological spaces satisfying condition (M).
When discussing the Orlicz spaces (for instance, [5]) we introduce a certain functional having the properties which suggest the definition of a more general notion called a pseudomodular (in particular a modular). There are some versions (for example, in spaces with or without a partial order) and special cases of this notion ([9], [10] etc.). We shall use linear spaces without an order. Using a pseudomodular, we can define a modular convergence. In general, this convergence has no topological character.
A pseudomodular generates a filter which has a countable base. This filter is like the neighbourhood filter of zero in a linear-topological space. The filters (and their bases) generated by pseudomodulars have the algebraical and set- theoretical properties which are needed in definitions of new notions called a modular base ([6], [7]) and a modular filter. A modular filter needs not have any countable bases. Further generalized modular spaces are defined. In modular spaces F-pseudonorms or pseudonorms are introduced and, analogously, in the generalized modular spaces certain linear topologies can be constructed. The modular convergence can have no topological character but in fundamental cases ([6], [7] and [ 8] with certain changes) it has a pseudotopological description and the corresponding spaces may be regarded as special cases of linear-pseudotopological spaces.
1. Pseudotopological spaces. Let A be a non-empty set and let 2х = \A: А а X ].
1.1. A non-void family & с I х is called a filter in X if the following conditions are satisfied:
0 ф
if F e ^ , F a G с X, then G e J * ; if F, G g SF, then F n G e^F.
Here the symbol 0 denotes the empty set.
1.2. A non-void family с: 2X is called a. filter-base in X if the following conditions are satisfied:
0ф @ \
if A, Be3$, then there exists a set C e $ such that C c z A n B .
A filter in X is a filter-base in X (and in every Y z> X). If $ is a filter- base in X, then the family
\0\ = {A a X\ there exists a set B e $ such that A z> B} = \0\x is a filter in X. If 3F = [ # ] , then we say that ^ is a base o f the filter A filter [{Л }] (defined in a set X), where i c i , А Ф 0 , will be denoted by [A ], a filter [{ x }], where x e X — by [x].
1 3 . The set F (X ) of all filters in X # 0 can be partially ordered by the inclusion. Then for every family { ^ j } jeJ <= F (X ) (where J Ф 0 ) inf J^ -e F (X )
JeJ exists and
i n f j = П & j = {U Fj- F j e & j , j e J } .
jeJ jeJ jeJ
Moreover, if the family { satisfies the condition: for arbitrary sets F u . . . , F „ e (J & j (n is a positive integer) the set F x n ... n F n is non-empty,
jeJ
then sup ^ j e F ( X ) exists and
jeJ
sup = { F j n ... n F „ : n e IS, F u . . . , F „ e (J &f\,
jeJ jeJ
where IS denotes the set of all positive integers.
1.4. Let non-empty sets JC, Y and a mapping /: X -* Y be given. Then for a filter ^ e F ( X ) the filter
/ ( Я = [ { / ( 0 : F e P } - ] can be defined in Y (/ (F) = {/ (x): x e F } ) .
1 3 . The mapping т: X -»• 2FW is called a pseudotopology in X ([1], [2], [3]) if the following conditions are satisfied:
if F ( X ) 3 # r z> У ет (х), then ^ е т ( х ) ; if J 5", ^e t(x), then п ^ е т ( х ) ; [x ] e T (x)
for every x e X .
Pseudotopologies fo r modular spaces 191 If t is a pseudotopology in X , then the pair (X , x) (also X ) is called a pseudotopological space. The filters from x (x) are called convergent to x in the space (X , x) (or with respect to the pseudotopology x).
1.6. Let f rx be the neighbourhood filter of x e X for a topology in X and let
x r (x) = { 3 F e F ( X ): & =э У/ J , where x e X . Then тг is a pseudotopology in X.
If a is a pseudotopology in X and there exists a topology ЗГ in X with x? = <7, then we say that о is a topology. We shall identify the topology with the pseudotopology x^. Hence, one may write x? instead of ZT and (X , z r ) instead of (X , ZT).
The condition
infx(x)Gx(x), x gX , is necessary for the space (X , x) to be topological.
1.7. Let (X , o), ( Y, x) be pseudotopological spaces. We say that a mapping/: X -> Y (or /: (X , <т)->(У, x)) is continuous at a point x e X if for every .<Feo(x) the filter / (#") belongs to x(/(x)).
For topological spaces this definition is equivalent to the classical one.
If a mapping/: (X , <т)->(У, x) is continuous at every x e X , then we say that / is continuous (in X , in (X , a)).
1.8. The set P (X ) of all pseudotopologies in X can be partially ordered in the following way:
or ^ x if and only if <т(х)=эх(х) for every x e X ; o ,x e P ( X ) . For <7, x e P (X ), a ^ x, one may write (X , o) < ( * , t).
Let x 2 be topologies in X . Then < F 2 (or x? < xr2 ) means that .fTi c= 3~2.
The Cartesian product of pseudotopological spaces is defined similarly as for topological spaces.
1.9. Let (X , x) be a pseudotopological space. We say that an MS-sequence S = (Xj)Jej consisting of elements of X is convergent to x e X with respect to the pseudotopology x (or in a space (X , x)) if the filter
& (S ) = [{{xj-: j ^ jo ] : j 0 € J } ] belongs to x(x).
2. Linear-pseudotopological spaces. Let К be the field of real or complex numbers and let X be a linear space over K . We shall use the following notations :
16 = {X e K : |A| ^ <5}, where 3 ^ 0 ;
V — the neighbourhood filter of O e K (the field К will be equipped with the usual topology);
A ± B = { a ± b: a e A , b e B ], where A , B c : X ; A ± b = A ± {b}, where A c= X, b e X ;
A -A = {Aa: A eA , a e A ] , where A <= K , A cz'X;
A -a = A- [a], where A a K , a e X ; A-A = {A}-A, where A e K , A <= X ;
± & 2 — ± F 2: F l e ^ r1, F 2e & 2} l where .^ x, ^ 2e F { X ) ; .¥ ± x — ,(F '± [x ], where e F ( I ) , x e X ‘,
^ - . F = [{ G -F : G e'S , F e& \ \ , where % e F {K ), ? seF {X )\
A • = [Л ] • ,^r, where 0 ^ Л с K , ^ F e F ( X );
A'& = [A ]-.? , where A e K , :W eF (X );
f4 - x — r4 ' [ x ] , where ^ e F ( K ) , x e X .
2.1. We say that т e P { X ) is a /mear pseudotopology in a linear space X ([1], [2], [4]) if the algebraic operations
+ : X x X - ^ X , ■ : K x X - * X
are continuous (where the space X is equipped with the pseudotopology x). If г is a linear pseudotopology in X, then the pair (X , i) is called a linear- pseudotopological space.
Every linear pseudotopology т in a linear space X satisfies the following conditions:
(oc) if F (X )э .^ г э ^ет(О ), then ^ ет (О );
(P) if .F , ^ет(О ), then n ^ e т(0);
(1) if e x(0), then # Ч -^ е т (0 );
(2) if A e K , ^Fex(O), then A- J^ er(O );
(3) if .^ 6 1 (0 ), then Е -^ е т (О );
(4) if x e X , then V -x ex (0 );
t(x) = т(0) + х = \.W + x\ .^ e x {0 )] for x e X .
2.2. If a set t(0) c: F (X ) (where X is a linear space) satisfies conditions (a), (P), (1H 4) and if т (x) = т (0) + x for x e X , then т is a linear pseudotopology in X.
2.3. Let u, x be linear pseudotopologies in X. Then a ^ x if and only if a (0) о г (0).
2.4. A linear pseudotopology т is a linear topology if and only if in fi(0 )E i(0 ) (cf. with 1.6).
3. Balanced and equable linear-pseudotopological spaces. Let X be a real or complex linear space.
Pseudotopologies fo r modular spaces 193
3.1. We say that a set А с X is balanced if bal А = IA = A.
3.2. Let us observe that:
if A = ba\A, X ç K , then XA = bal(/./l),
if A = baM , X, f i e K , \X\ ^ \p\, then ХА а цА.
3 3 . We say that a filter # in a linear space is balanced if bal F = I F = F .
Let us observe that for every filter F in a. linear space the condition I F c= F holds.
3.4. If a filter F is balanced, X, /i gK, |A| ^ \p\, then с X F (cf.
with 3.2).
3.5. A linear pseudotopology т in a linear space X (and a space (X , i)) is balanced ([4]) if the following condition is satisfied:
for every J* e T (0 ) there exists а ^ет(О ) such that F з> ^ = VS (an equivalent condition: if F e x(O), then I F et(0)).
3.6. For every filter F in a linear space the filter I F is contained in VF. So, we have the following theorem:
L et a set t(0) c= F (X ) (where X is a linear space) satisfy conditions (oc), (p), (1), (2), (4) from 2.1 and the condition:
(3b) i f F e r(0), then I F et(0).
M oreover, let т (x) = т (0) + x fo r x e X . Then x is a balanced linear pseudotopology in X.
3.7. We say that a filter F in a linear space is equable if V F = F ([2], [4]).
3.8. We say that a linear pseudotopology i in a linear space X is equable (and that a space (X , x) is equable) if for every .г^ет(О) there exists a filter ^ et(O ) such that F з = VS ([2], [4]).
3.9. Every equable filter is balanced; hence an equable linear - pseudotopological space is balanced.
The neighbourhood filter of zero in a linear-topological space is equable, so a linear-topological space is equable. There exist equable linear- pseudotopological spaces which are not linear-topological.
3.10. For a linear pseudotopology x in a linear space X there exists a (unique) equable linear pseudotopology т * ^ т such that for every equable linear pseudotopology a ^ x the condition x # ^ a holds. Moreover, we have
r # (0 ) = \ FeF (X ): there exists a filter ^ 6 1 (0 ) with F :э V4).
4. Generalized modular spaces. Let X be a linear space over the field К of real or complex numbers.
4.1. We say that a non-empty family 0b c= I х is a modular base in X ([6], [7]) if the following conditions are satisfied:
(M l) for every sets [/l9 U2 e^b there exists a set U e & such that r ( U )
= (ax + fiy\ x, y e U, |a| + |/*l ^ 1} a U 1 n U2,
(М2) every set U e ^ is absorbent in X (i.e., for each x e X there exists a number а Ф 0 such that a x e U).
A modular base in X is a filter-base in X. If M is a modular base in X , then [J>] is called a modular filter in X . Every base of a modular filter in X is a modular base in X .
4.2. Let M (X ) be the set of all modular bases in X . We define in M (X ) the following relations:
0bx < 0b2 if and only if there exists a number a / 0 such that [0bx] c a [ # 2],
0 b x ~ 0 b 2 if and only if there exist numbers a ls a 2 Ф 0 such that
«1 [ ^ 2] 01 [ ^ 1] «2 [ ^ 2] ^ and only if 0bx < 0b2 and 0b2 < 0 b x)\
0b!, 3b2eM (X ).
The relation -< is reflexive and transitive, ~ is an equivalence relation.
Let us observe that 0b ~ [J* ] for every modular base J*.
4 3 . Let J ’gM (X ) be given. We define 0b~ = { e M( X ) :
If I eM (X ), then the pair (X , ^ ~ ) is called a generalized modular space (shortly: a modular space).
4.4. Let us define in a generalized modular space a linear pseudotopology. We have the following theorem:
I f ( X ,0 b ~ ) is a generalized modular space, 0b x e 0b~, x(0) = { F gF (X ): there exists a À e K such that 0F zz>
x(x) = x(0) + x fo r x e X ,
then t is a balanced linear pseudotopology in X ; moreover, all bases from 0b~
generate the same pseudotopology x.
4.5. In [6] a convergence of M S-sequences in modular spaces is defined in the following way: An MS-sequence S = (Xj)jeJ consisting of elements of a space X is called convergent to x e X with respect to the modular base 0b in X if there is a number а Ф 0 such that for every U e0 b there exists a j 0 e J such that for every j e J , j ^ j 0, the condition a ( x ,— x )e U holds. Moreover, if 0bx ~ 0b, then S converges to x with respect to 0b if and only if S converges to x with respect to 0bj . So we introduce the definition: An MS-sequence S is convergent to x in a space (X , 0b~) if S converges to x with respect to 0b (or with respect to any base 0bx^0b~).
Pseudotopologies fo r modular spaces 195 We have constructed a pseudotopology in a modular space. Now let us observe that the above definition of a convergence of MS-sequences is equivalent to the one given in 1.9.
In the sequel a generalized modular space {X , 08~) will be identified with the linear-pseudotopological space (X , t) with т defined as in 4.4.
5. Condition (M). Let X be a linear space over the filed К of real or complex numbers.
5.1. We say that a linear pseudotopology т in X (and a space (X , t)) satisfies condition (M) if there exists a filter J^ ei(O ) such that for every
^ет(О ) the condition ^ n> AF holds with a certain AeK .
5.1.1. R e m a rk . The filter F from the above condition is a subset of the filter [0].
Indeed, the case X = {0} is trivial. So, let X Ф {0}. We take an x eX, x Ф 0. Исет(О), so there exists a A e K such that Vx zdA F . The case A = 0 implies that Ух zd[0]. So {0} => I e • x for a certain e > 0. But it is impossible.
Therefore А Ф 0 and Vx id F . Moreover, [0] id Vx. So [0 ] => F . 5.1.2. R e m a rk . In condition (M) we may assume that А Ф 0.
Indeed, if A = 0, then we obtain ^ id[0 ]; so ФА = [0]. But [0 ] => F
— I F (Remark 5.1.1).
5.2. Let us observe that a generalized modular space is a balanced linear-pseudotopological space satisfying condition (M).
5 3 . Let (X , t) be a balanced linear-pseudotopological space satisfying condition (M). We have
(M) there exists a filter F e z ( 0 ) such that for every ^ет(О ) there is a А Ф 0 for which У id A F .
5.3.1. R e m a rk . The filter F need not be balanced.
Example. X = R over R (the set of all real numbers), F
= [< —1, 2>] (<a, b } denotes a closed segment).
F need not be balanced but in (M) the filter I F may be taken instead of F . Indeed, we have F zd I F . So, if ^ id A F , then ^ о AI F . Moreover, there is а ц Ф 0 such that I F id f i F . Hence ^ id A I F implies that ^ => AptF.
5.3.2. Theorem. L et (X , x) be a balanced linear-pseudotopological space satisfying (M) with F . Then the follow ing conditions hold:
(bM l) there exists a p Ф 0 such that I F + I F id p F , (bM2) Vx F f o r every x e X .
P r o o f. The filter I F et(0). So I F + I F et(0). Therefore I F + I F
zdp F , where p is a certain number different from 0. Now let an x e X be given. Ехет(О), so there is а А Ф 0 such that Vx id A F . It means that Vx
ZD F .
5.3.3. Theorem. L et F be a filter in a linear space X. M oreover, let F satisfy conditions (bM l) and (bM2). Then the equalities
t(0) = { S e F i X ) : S з k F for a certain к Ф 0],
t(x) = t(0) + x (where x e X )
define a balanced linear pseudotopology in X satisfying condition (M).
P ro o f. We shall apply Theorem 3.6. Conditions (a), (2), (4) are obvious.
We consider conditions (1) and ((3). Let filters S x, S 2 e z (0 ) be given. Then there exist k { , k 2 Ф 0 such that <S l з k x F , S 2 3 ^2 ^ • For ^ = 1,2 we have S k ^>kk F z i k k l F ^(\kfi + \k2\ )-IF (see 3.4). So ^ + S 2 з Ш + \k2\) x
x ( I F + I F ) з {\k1\ + \k2\)p^. Therefore the filter ^ i + C#2 belongs to i(0).
Moreover, we have fS x c\(§ 2 => (l^il + ^ 2! ) = > (l^-il + |^2|) ( 1 F + I F ) з (|ЯХ|
+ \k2\)p.^. So ^ п ^ 2 бт(0).
(3b): Let F (X ) 3*S з k F , к Ф 0. Since VS з k L F з к ( I F + I F ) з k p F , the filter VS belongs to i(0). Obviously, the pseudotopology т satisfies condition (M).
5.3.4. R e m a rk . Condition (bM l) is equivalent to the conjunction of the following ones:
(lb M l) there is а к Ф 0 such that + d I I -, (2bM l) there is a Ç Ф 0 such that V F з ^ F .
Indeed, from these conditions we have I F + I F £ ,F + £ ,F = £ ,(F + + .F) 3 E,kF, from (b M l) we get: F + F I F + I F zs p F , I F zdI F + I F
3 p F .
Moreover, if condition (2bM l) is satisfied, then condition (bM2) is equivalent to the following one:
every set F e F is absorbent in X.
First, let condition (bM2) be satisfied and let arbitrary F e F , x e X be given. We have Vx з F . So there exists an f, > 0 such that F з I e x. Thus e x e F (condition (2bM l) has not been applied). Now, let each F e F be absorbent in X and let the filter F satisfy condition (2bM l). For every x e X , F e ^ there are G e F , a # 0 such that f з - i G and ocxeG. So F 3 - J G
£ £
a
3 — lx e Vx.
£
5.3.5. R e m a rk . In [ 8] a certain notion called a premodular base is considered. Let S be a (real) linear lattice. We say that a filter-base 2S 3 2s is a premodular base if the following condition is satisfied:
(c) there exists a ft Ф 0 such that for every set U e M there is a set U 'eM with fi(N (U ') + N(U')) a U, where N (U ’) = { y e S : \y\ < |x| for a certain x e U nr
Pseudotopologies fo r modular spaces 197 Let us observe that condition (bM l) can be written in the following way (real and complex cases):
there exists a such that for each U e F there is a U'gF with P{IU' + IU') cz U.
This condition is like condition (c). Here the set W is instead of N (U ’).
5.4. Let us give some theorems on spaces satisfying condition (M).
5.4.1. Theorem. A linear-pseudotopological space satisfying condition (M) is linear-topological if and only if it is equable.
P ro o f. The necessity is obvious (3.9). Let us verify the sufficiency. Let (X , t) be the considered equable space and let F gt(0) be the fixed filter appearing in condition (M). Since the pseudotopology т is equable, a filter (S X gt(0) can be chosen such that F гз V^SX. Moreover, there exists a v ^ O such that Ve# x => vF . So F =э V^x = - VrS^ => F , and hence F = Ve# ^. Thus
v
for each X Ф 0 the equality X F — F holds. It means that t(0) = \&eF{X): &z>F} .
So, by virtue of 2.4, our space is linear-topological.
5.4.2. Corollary. A generlized modular space is linear-topological if and only i f it is equable.
5.4.3. Theorem. L et a linear-pseudotopological space (X, t) satisfy condition (M) with a filter F . T he space (X , i) is linear-topological i f and only if there exist numbers p, v # 0 such that \p\ < |v| and p F c= \ F .
P ro o f. The necessity is obvious because if (X , i) is linear-topological, then the filter .F is equable, and for every X Ф 0 the condition X F = F is satisfied. Now we verify the sufficiency. The conditions p F c= v F , p, v Ф 0,
|^| < |v| imply the following ones: F <~ — F , \v/p\ > 1. The filter V Fv h
converges to 0, so V F =» F . Now let us take an arbitrary set A e VF. Then A => eIF for some e > 0, F e F . \v/p\ > I, so there is a positive integer n such that E\v/p\n > 1. Since the filter F is a subset of — F , F => v (v/p)nF' for a
F
certain set F ' e F . Therefore A zdeI F з E{v/p)nIF' => F'. Hence AgF . It means that V F c F . Moreover, F c VF> so F = VF. Therefore our space is linear-topological.
5.4.4. Corollary. L et a linear-pseudotopological space (X , z) satisfy condition (M) with a filter F . Then the space (X , r) is linear-topological if and only i f p F = v F fo r som e p, v Ф 0 with |/i| Ф |v|.
5.4.5. Theorem. L et a linear-pseudotopological space (X , i) satisfy condition (M). Then there exists a (unique) linear topology t v ^ т such that fo r every linear topology F ^ т the condition i v ^ F holds. M oreover, t v = i #.
P r o o f. It is obvious that t# ^ t (3.10). Let F be the fixed filter of condition (M) and let an arbitrary ^ е т # (0) be given. Then there exists a filter (0) such that У з . Moreover, ^ о X F for some X Ф 0. So ^ з з XVF = VF. Furthermore, the conditions Ж e F (X ), Ж з VF imply Ж еt#(0), so
t# (0) = {& e F { X ) : ^ з V F }.
Therefore i # is a linear topology in X. Now let an arbitrary linear topology Ж ^ t be given. The topology Ж is equable (3.9), so we get Ж ^ x # (3.10).
The uniqueness of the topology t v is obvious.
5.4.6. R e m a rk . There exist linear pseudotopologies т for which there is no linear topology r v ^ т satisfying the following condition:
if a linear topology Ж ^ t, then Ж ^ t v .
Example. X is an infinite-dimensional linear space,
t(0) = { F e F ( X ) : F 3 Vxx + ... + Vxn for certain x lf . . . , x ne X } ,
t(x) = t(0) + x for x e X .
Here there is no linear topology which would be ^ t. Let us observe that т #
= г and t # is not a linear topology.
5.4.7. R e m a rk . Let a balanced linear pseudotopology т (in a certain linear space over the field К ) satisfy condition (M) with a filter F . Then sup {eF : e > 0} is the neighbourhood filter of 0 for the topology t v = t*, i.e.
VF = sup {eF : e > 0}.
Moreover, if e„ Ф 0, en -> 0 as n -* oo, then V F = sup {e„ F\ n e N } . Indeed, we have
V F = [_{eIF: e > 0, F e F}~\ = sup {eI F : e > 0}.
Since there exists a fi > 0 such that I F <= F <= \ iIF , we get sup [eI F : e > 0} 3 sup {eF : £ > 0} c sup [e^lI F : 8 > 0}
= sup {eI F : e > 0}.
So sup {eF : £ > 0} = V F (see also 3.1 in [6]).
Now let an £ > 0 be given. F or an e„0 with |£„J < Efi we have eF cz e^ iIF <= e„0 F c sup {e„ F : n e N } . Therefore sup {eF : e > 0}
c= s u p {£ „ ^ : n e N } . Moreover, from e„ F c: En^ I F <=■ \E„fi\ F (n e N) we get sup {e„ F : n e N} cz sup [eF : £ > 0}. Therefore supjfiJ^: e > 0}
= sup {e„ F : n e N } .
Pseudotopologies fo r modular spaces 199
5.4.8. Theorem. L et (X , i) be a linear-pseudotopological space. Then there exists a (unique) linear topology т л ^ т such that fo r every linear topology < I the condition f f ^ т л holds.
P ro o f. From [4 ] we get that
sup {& e F ( X ) : & c z % n (& + & )n V & },
where Щ1 = infr(O), is the neighbourhood filter of 0 for the topology т л. Here we shall give another construction of this filter (see [6]). It is known that T ~ eF (X ) is the neighbourhood filter of 0 for a linear topology in X if and only if the following conditions are satisfied:
(LT1) f/~ <zz T" -f- f (LT2) f c / f ,
(LT3) 'V' c Vx for every x e X .
Let ^ = infx(0) and У = [{ (J H i + • .. + Л„): A u A2, . . . e W } ] . It is easy
n = 1
to see that 'V c W . We shall show that У is the neighbourhood filter of 0 for a linear topology in X. Let an arbitrary x e l be given. We have Vx zd%
d / f c f , so condition (LT3) is satisfied. Now let a set Ve У be given.
00
Then there exist U ly U2,...e< % such that V zd (J (I U l + ... + I U n). By
П— 1
virtue of the equality
u
( I U .+ ...iu„) = /• U (K/i+ ... +iun)
n= 1 n= 1
we infer that V eli^ . So condition (LT2) is satisfied for the considered filter.
On the other hand, for the same sets V, U lt U2, ... we get
V=> (J ( Я / . + . . . + Я / „ ) = . G ( Н и ^ и 2) + ... + I ( V 2k_ l n U 2t)) +
n= 1 J t = l
00
+ U ( l( U i n U 2)+ ... + l ( U 2t- 1 n U 2t) ) e r + Г ,
k = 1
so condition (LT1) is satisfied. Let us denote by & 2 the linear topology for which У is the neighbourhood filter of 0. Since У cz I <zz <%, x < t. Now
let W be the neighbourhood filter of 0 for a linear topology in X and let
;T ^ t. Obviously, we have i f c t and = W . We shall consider the filter
<» = [{[) W + ••• + и у : Hi, ...е тГ }].
л= 1
The condition iV а I % implies W а V . Let a set W e iV be given. Since iV cz'W'+'W, there exist sets Wu W2, . . . e i t r satisfying the conditions WzdW1 + W1, Wt zdW2 + W2, . . . , W„_1 z> Wn+W„, . . : Therefore we have
Wt + ... + W n_ 1 + W „czW 1 + ... + W n- 1 + W„+W„ŒW1+ ... + w n. 1 + 00
+ И^_1 c: ... c + Wl cz W. Thus W zd (J (W±+ ... + tT „ )e ^ . Therefore n— 1
We &. Hence ii a (moreover, <& c 'W\ so W — °Ж). Now we can see that # ' c= У . It means that T ^ J r 1. So ^ = т л.
5.4.9. Let a linear pseudotopology т satisfy condition (M). Then we say that t v is the Orlicz topology for the pseudotopology т and that т л is the Wiweger topology for т (see [8]).
References
[1] H. R. F is c h e r, Limesraume, Math. Ann. 137 (1959), 269-303.
[2] 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).
[3] W. G â h le r, Grundstrukturen der Analysis I, Berlin 1977.
[4] —, Grundstrukturen der Analysis II, Berlin 1978.
[5] M. A. K r a s n o s e l’s k it and Ya. B. R utickiT, Convex functions and Orlicz spaces, Groningen 1961.
[6] R. L esn iew icz, On generalized modular spaces I, Comment. Math. 18 (1975), 223-242.
[7] —, On generalized modular spaces II, ibidem 18 (1975), 243-271.
[8] —, and W. O rlic z , A Note on Modular Spaces. XIV, Bull. Acad. Polon. Sci., Sér. Sci.
Math. Astronom. Phys. 22 (1974), 915-923.
[9] J. M u sie la k and W. O rlicz , On modular spaces, Studia Math. 18 (1959), 49-65.
[10] H. N ak an o , Generalized modular spaces, ibidem 31 (1968), 439-449.
