Mixed non-archimedean topologiesAbstract.

ROCZNIKI POLSK1EGO TOWARZYSTWA M AT EM AT YCZN EGO Séria I: PRACE M ATE MAT YCZN E XXII (1981)

Monika Fabijanczyk (Lôdz)

Mixed non-archimedean topologies

Abstract. In the present paper there has been described a theory of the mixed topology and two-norm convergence in the case of a topological vector non-archimedean space (i.e. such a space that possesses a base of neighbourhoods of в, composed of sets U such that U + U a U ). The principal results o f this theory in real locally convex spaces hold true in the case of non-archimedean spaces, too. The new important result is the fact of the validity of an equivalent of Banach-Steinhaus theorem on a convergent sequence of linear and continuous maps in mixed topology. In the case of real spaces, in general, this theorem is false.

P A R T I

The purpose of this paper is to develop properties of a two-norm convergence and mixed topology in a case of non-archimedean linear topolo­

gical spaces.

A linear topological space over a valuated field Ж' is called a non- archimedean (n.a.) space if it possesses a base of neighbourhoods of в whose all elements U satisfy the following condition:

(*) U + U c= u.

A set S c= E is said to be Ж -convex if ax + fiy e S for all pairs of scalars а,/?е.Ж satisfying the condition |a| ^ 1, |/?| ^ 1. A linear topological space over a valuated field Jf is a locally Ж -convex space if it possesses a base of neighbourhoods of 6, composed of Ж-convex sets.

It is possible to prove:

Theorem 1.1. Let E be a linear topological space over an n.a. valuated field Ж . E is a locally Ж -convex space if and only if it is an n.a. space.

The necessity of the theorem is proved by the observation that for any Ж -convex set S,S + S <= 5 holds.

P r o o f o f sufficiency. Let A be a base having properties described in the definition of an n.a. space. For a given U e.A , the set Ü will denote the kernel of balance of U. Since in all linear topological spaces over an

arbitrary non-trivial valuated field there exists a base of neighbourhoods of в, consisting of balanced sets, we have:

Ü Ф 0 for any U e M .

Hence, considering that an algebraic sum of any two balanced sets is balanced and Ü + Ü a U + U c U ; Ue/â, we see that:

0 + 0 C 0, UeJ0.

Then M = {Ü : U e M } is the base we were looking for. ■ Let us adopt the following notations:

(a) Ж will denote an arbitrary n.a. non-trivial valuated field.

(b) Let г be an n.a. topology in a linear space E over a field Ж . JT-nbh(t) stands for the set of all JT-convex т-neighbourhoods of в. Without loss of generality of our considerations we can assume that

(i) for any sets С/1? (У2е ^ (т ) there is a set и ъе М {т) such that U 3 a U 1n U 2,

(ii) if U e / i{t), then for any А еЖ and Я Ф 0, A U eM (x).

(c) E, p, s/ will denote respectively a linear space over a field Ж , a locally Ж -convex Hausdorff topology in E, a non-empty family of JF-convex subsets of E. ■

Theorem 1.2. Let a family Ж satisfy the condition:

(I i) for any set Ae.pJ and non-zero scalar Х еЖ there is a set A 'esJ such that A c= Xsé'.

Then there is the finest locally Ж -convex topology p , in E such that all identity maps:

iA : A -» E ; A e Ж

are continuous, where A is provided with the topology p\A. The family S i p f i of all Ж -convex and absorbing subsets of E, such that for each and each A es/ there is a set V*€/#(p) which satisfies the condition V a V * n A , forms the base of pr/-neighbourhoods of в.

P ro of. First, it must be shown that the family in agreement with a well-known theorem of the general theory of linear topological spaces, defines a linear topology p,, on E, for which it is the base of neighbourhoods of 6. From the definition of this family it follows that we must only show that Ve/S{pr/), then / V e .^ ip fi for any 0 # к . / '. It is easy to see that it sullices to consider the case of \/\ < 1.

For each А е Ж we can choose a set A 'es/ such that A a A A' and a neighbourhood V f e M (p) such that V => V f n A'. Since the map

x -* — x1 Я

of the space {А, ц\А) in (E ,p ) is continuous in 0, so for А ел/ there exists W*eâ${n) which satisfies the condition Wjf n А сz XV* . Consequently,

W f n A c (XVf.) n A c (XVJf) n {ХА') = X (Vf- глА') a XV

for А е л / . Thus XVe^(p^). And so, we observe that pi^ is the finest locally Jf-convex topology for which the identity map defined in the statement of the theorem are continuous whence p ^ p^. As a result, p^ is a Hausdorff topology. Since the sets of the family 0H (p^) are Jf-convex, the topology is locally JT-convex. We shall call a mixed topology the above-mentioned topology. ■

Theorem 1.3. Let the assumptions of the preceding theorem be satisfied.

Then:

(a) /v|A = p\A<for each А ел / .

(b) Let (F, t) be a locally Ж -convex space. The linear map f : (E, p^) -> (F, t) is continuous if and only if the map f\ A : (A, p\A) -> (F, t) is continuous for each Аел/

(c) I f v is a locally Ж -convex topology in E such that p\A = v|A: Аел/, then pp/ = Vtf.

(d) pQ/ = (//,,),✓ •

(e) A non-archimedean seminorm p on E is px/-continuous if and only if the restriction p\A is p\A-continuous for each Аел/.

(f) The topology p^ is coarser than any localy Ж -convex topology pi' on E which has the property described in statement (b) with the replacement of p* by p'.

P roof.

(a) This follows from Theorem 1.3 and the fact that (A, p^\A) and (A,p\A) are additive topological groups.

(b) The necessity of the condition follows directly from the identity:

f\A = f o i A; Аел/.

Proof of sufficiency. Let U be an arbitrary JT-convex i-neighbourhood of в. Then / _1([7) is non-empty JT-convex and absorbing subset of E and f ~ 1(U )n A = (f\ A )~ x(U) for each А е л / . It follows from the assumption of the theorem that / - {(J) n A is a //^-neighbourhood of 0. Consequently, there are V f e ^ ( p ) such that /~‘ (U ) з A n V f for A es/. Therefore f 1 (U )e & (p tf). .

(c) This follows from (b).

(d) This follows from (a) and (c).

(e) The necessity is proved by the observation that p\A = p o iA; А ел/.

Proof of sufficiency. Let т be the locally Jf-convex topology in E

determined by the one-element family of seminorms {p}. Then it follows from the assumption of the theorem and condition (b) that the identity map i : (E, p^) -*■ (E, i) is continuous. Therefore p is p^-continuous.

(f) Since the identity map i : (E, p^) -> (E, p^) is continuous, the restriction map i\A: (E , p\A) -> (E, /v) is also continuous for each A es/. As a result, for any locally Ж -convex topology described in the assumption of this theorem, the map: i: (E, p') -*■ (E ,p *) must be continuous. Hence p' ^ pr/. ■

We shall assume in the sequel that (I2) и A = E.

Theorem 1.4. I f s/ satisfies conditions (1ц), (I2), then the collection of sets

(B2) Jf-conv( U (A n V f)),

where V f ranges over the family Ж (p), forms a base of Ж -convex prV-neighbour­

hoods of в.

P roof. Let F be an arbitrary set of the form (B2). It is jF-convex. We have to prove that it is also absorbing. Let x e E . It follows from condition (I2) that x gA 0 for some A 0esZ. The set VfQ is absorbing so there is a real number a ^ 1 such that xeX Vf0 for all Х е Ж , |A| ^ a. Hence

x e A (A 0 n Vff) cz X JF-conv ( (J (A n V f)) = XV for AeJF, |A| ^ a,

0 АЫ

and thus F is an absorbing set.

Next, we observe that since V ^ > A n V f ’, A es/, so Fe J T -n g b ^ ).

Finally, we shall show that for each neighbourhood U of 9 it is possible to find some set of the form (B2), which is contained in U. It follows from Theorem 1.3 (a) that the set U n A is a /^-neighbourhood of в for each A es/, and consequently, U n А zd V f n A for some V f e ^ ( p ) . Hence

U ( A n V f ) a U (A n U) = U.

Ae,<# Ae.<xf

Since U is JF-convex, so

jF-conv ( (J (A n V f)) cz U. •

Ae.<?/

Let’ J t , J f be two non-empty families of a subset of a set H. We say that the family J f maximizes the family M if, for any set M e l , there is a set N e Ж such that M cz N .

It the families M and J f maximize each other, then we say that they are cofinal.

Theorem 1.5. Let s/y& be non-empty families of Ж -convex subsets of the space E. l f s / , & satisfy condition (Ij) and $ maximizes s/, then

P roof. It is enough to prove that if Ve Ж -nbh{ps), then Ve Ж-пЪЩр^).

Let us fix an arbitrary set A e s é . Since 0b maximizes sé, so A a В for some Be 01. It follows from Theorem 1.2 that there exists a set Vfeâb(p) such that FB* n В c= V. Hence

V f n A c V f n В c F.

Since this holds for each set Т е л / , therefore FeJf-nbh (p^). ■

Corollary 1.1. Let the assumptions of the preceding theorem be satisfied.

I f the families л/ and 0& are cofinal, then Me/ M^ •

Theorem 1.6. I f sé satisfies conditions (1Х), (I2) and there exists a set A0esé such that Int A0 ф 0 , then px/ = p.

Proof. It follows from Theorems 1.2 and 1.4 that it is enough to prove that any set of the form (B2) is a //-neighbourhood of в. But this is trivial because

Ж -conv (J (A n V f) z> V/ n A0 and V f n A0e Ж -nbh (p). ■

Лес/ u u

Theorem 1.7. Let sé satisfy condition (IJ. p^ is the finest locally Ж -convex topology in E coinciding with p on all A e s é .

P ro of. Let г be a locally Ж -convex topology in E coinciding with p on all A esé. We shall prove that the identity map: i : (E, p^) -+ (E, t) is continuous.

It follows from the assumptions of this theorem that the maps i|T: (E, p\A) (E, t) are continuous. Hence by Theorem 1.3 (b), the theorem is concluded. ■

In order to obtain any useful results, it is necessary to impose the following restriction:

(I3) For any sets A t , A 2esé there exists a set A 3esé such that A t + T 2 c= A 3.

It is easy to see that each family of -convex subsets of E, totally ordered by inclusion, satisfies this condition.

Theorem 1.8. I f conditions (I^ , (I2), (I3) hold, then

M</ Me/ ’ where sé : = {Л м: A e sé }.

We shall first prove the following lemma:

Lemma 1.1. Let S be а Ж -convex subset of E and p — an n.a. seminorm on E. A functional p\S is equicontinuous if and only if it is continuous in в.

Indeed, the above follows from the fact that S is an additive topological group.

P r o o f o f the theorem. A family sé maximizes sé and hence, by Theorem 1.4, we get that

We shall prove that if any n.a. seminorm p is p^-continuous, then it is Pj-continuous. Then p# ^ p^ (see [9]). Let A be an arbitrary set of the family s é . Since, by Theorem 1.3 (e) and Lemma 1.1 functional p\A is /^-continuous, therefore it possesses a unique extension pA on Â, being a p\Л -uniformly continuous map. We have finally proved that pA = p\Â.

Then, by Theorem 1.3 (e), p is /^-continuous. Indeed, let x e A . Then there exists an M - S -sequence (xt),6T of points of A, p-convergent to x. From condition (I2) it follows that x e A ' for some A ’ e s é . We find, by (I3), a set A " esé such that A + A 'cz A". Hence x e A ", and since p\A” is /^"-continuous, so (p(xt))teT is convergent to p(x).

On the other hand, (p(xt))teT is convergent to pA(x). Hence p(x) = pA(x) (E is a Hausdorff space). Since this holds for each x e  , therefore p\ A — pA. ■

Corollary 1.2. Let the assumptions of the preceding theorem be satisfied.

I f a set M <= A for some A e sé, then M M = M “*

Indeed, since p^ ^ p, so A? is а ры -closed set. Consequently, f f l = M f,]À = = м "-*.

By the preceding theorem, we obtain М д = А?Дв/.

We shall now prove two theorems concerning equicontinuous families of maps.

Theorem 1.9. Let condition (IJ be satisfied, and let F be any locally X-convex space. A family ^ of linear maps from E into F is po/-equicon- tinuous if, and when condition (I2) holds, only if families ^ A : = {f\A: f e X ) are p\ A-equicontinuous at в for each A e s t .

P ro of. Suppose that & is an equicontinuous family of linear maps from E into F . Then, for any neighbourhood V of 9 in F, there is a JC-convex p^-neighbourhood U of 9, satisfying the condition

U c f - ' i V ) ; f e P .

For each A e sé we find, by Theorem 1.2 a U % e& (p) such that U => A n U*.

Consequently, ( / 1 A )~ x (F ) = А n / -1 (V ) => A n U* for / eJ *. Let us define

w t : = Г К / И Г 'О О ; A es/.

fe/s

It is easy to see that W* is а p | ^-neighbourhood of 0 and { f \ A) ~ 1( V ) = > Wf f for f e & .

Therefore ^ A is equicontinuous at 9.

P r o o f o f sufficiency. For each JT-convex neighbourhood of 9 by assumption, there are sets U%E&(p); A E d , such that

{ f\A) ~1(V)=> A n u%; f e 9 .

Let us observe that the set U := f ) f ~ 1(V) is Ж - convex. It is absorbing, fa?

too. Indeed, for each x e E , by condition (I2), we find a set A0E d such that х ь А 0. Since U% is a neighbourhood of 0, so x eXU^q; ХеЖ , |A| ^ a, where a ^ 1 is some real number. Therefore we get:

x e W%0nXA0 = Ц и Лоп А 0)

<= А(УМ)_1(Ю

= 1 ( Г ‘ ( П Ч | с / " ( 1 ' ) ; /e Jf,W > a .

Since this holds for each point x e E , so U is an absorbing set. Since U

=

n С Г Ч Ю п Л ) = П (

fa* fa*

for A E d . The theorem is proved. ■

The next theorem (theorem of Banach-Steinhaus for mixed topology) is false in a case of archimedean normed space.

Theorem 1.10. Suppose that E is a p-metrizable and p,/-sequentially complete n.a. space. Let conditions (IA), (I2), (I3) hold. I f a sequence (/„)*= t of p^-continuous linear maps from E into a locally Ж -convex space F is convergent to a map f 0 at each point x eE, then f 0 is linear and p^-continuous, and a family {/„}*=0 is p^-equicontinuous.

In the proof we shall make use of the following lemma:

Lemma 1.2 (see [1]). Let G be a complete metric space and (0„)®=i — a sequence of continuous maps from G to a metric space H. I f (g„)™=l is convergent in each point x of a set of the second category of Baire, then the family {#„}*= i is equicontinuous at each point of some set of the second category В c= G.

P r o o f o f theorem . It is easy to see that the map f 0 is linear. We can assume, by Theorem 1.8, that sets of the family d are ^-closed, therefore /^-closed, too. Let A be an arbitrary set of a family d . In the p IT-topology A is a complete metric space and the sequence (f n\A)™=l is convergent at each point of A to the map/0|A. By Lemma 1.2 we find such a point xa eA in which a family { f n\A}f= x is equicontinuous. Since

A is an additive topological group, so { f n\A}f=1 is ^ |/l-equicontinuous at в.

It follows directly from Theorem 1.9 that the family. {/„| Л }®=1 is equi- continuous in E. ■

P A R T II

In this section we consider a special case when •rf = { A } ”= i <s a denumerable increasing family of Ж -convex sets satisfying conditions (I !), (I2). It is easily seen that condition (I3) holds; then:

Theorem 2.1. All sets of the form

GO

(B3) V = V0* n П (V f + AÙ, where i = 0 ,1 ,2 ,...

i = 1

constitute a base of цг/-neighbourhoods of 6.

P ro of. Similarly as in the proof of Theorem 1.2, we shall first show that a family 2$ of all .Ж-convex oversets of sets of the form (B3) defines some locally Jf-convex topology on E, for which it is the base of neighbourhoods of 0.

Let any set U e .JA and ХеЖ ', X ф 0 be given. It follows directly from the definition of .JA that there is some V of the form (B3) such that U =э V.

It follows from the properties of Ж that there exists an increasing sequence of positive numbers (ik)k= i, satisfying the condition: Ak c= XAik for к — 1,2,...

We find U *, U *, ...e2â(n) in such a way that:

q - i ‘j + i -1

u t <= Я n v*, U f ^ x n V j* ;j = 1,2,3,...

i = 0 . i = i j and we get:

00

XU ^ XV = XV0* n П (XVf + XAi) i = 1

il-1 i2~ 1

= (XV0* n n (AK* + M-))n( n a V f + X A ^ n ...

i = 0 i = i j

n ( П ( X V f + X A f j n . . . id U$ ( ^ ( Щ + А ^ п . . . i=ij

n ( Uf + Aj) n ... n U* n П (Uf + Aj).

j = о Hence XUeJâ.

Next, it must be shown that U is an absorbing set. Indeed, let x e E . By condition (I2), x e A nQ for some n0. Let us choose a set such

n0- l

that U* cz V* п П V f - This set, as a neighbourhood of в, is absorbing, i = 1

and so x e W * , for AeJf,| A| ^ a, where a ^ 1 is some real number.

Consequently,

"0 - i

x e X ( U * n A „ 0) cz X(V0* n П Ш * п А Я0)

i = 1

c ^ n ' V p i ' + XJn П

i= 1 ‘ = "0

00

= Я(к0* п n № + 4 )) ■= ^

i = 1

for A e J f , |A| ^ a, whence U is an absorbing set.

As a result, ^ is a base of neighbourhoods of 9 for some locally JC-convex topology i on £ . Therefore all sets of the form (B3) also constitute a base of JT-convex neighbourhoods of 9 for the topology t.

We shall now prove that дсУ = t. Let F be an arbitrary set of the form (B3). For each n = 1 , 2 , 3 , . . . we find a set such that

n - 1

U* а П Vi*. Then

; = о

A„ n U* c A„ n F0* n ( П Vt*) n ( fl W* + Аг)) с A„ n V

i ~ 1 i = n

for n = 1, 2, 3,...

Hence, by Theorem 1.2, F is a ^/-neighbourhood of 9. In other words, T ^ /X,/.

To conclude the proof of this theorem, it must be shown that any set

00

U = Jf-conv 1J ( V * n A i ) , where V* e& (n ); i — 1 , 2 , 3 , . . . , is a i-neigh-

i = 1

bourhood of 9. Let (£/*)*= 0 be a sequence of elements of a base J'Gu), satisfying the conditions:

C*-i c= Fn*, U ï c l / ^ ; n = 1 , 2, 3, . . .

00

We shall now show that U =э F: = , (7Jn f) (С* + Лг).

i = 1

Let xeF; then x e U * and x = щ + щ, where а*еА,-, i = 1, 2, . . . Let as notice that

x = al + (a2- a 1) + ... +{an- a „ - 1) + un; n = 2, 3, . . . Since ax = х - щ е Щ - Щ cz Щ + Щ = U% a V? so Similarly :

= ип- 1 - и пеи*п- х + Щ - х = l/*-! с V*

4 — Roczniki P TM — Prace Matematyczne X X II

and

ая- а я- 1е А н^ А „ - 1 с А„

hence ап — ап _ х е V% п Ап for п — 2 ,3 ,...

On the other hand, by condition (I2), we find an index n0 ^ 2, such that x e A „Q. Therefore

un0 = x ~ an0e ^n0 — AnQ — A„q.

This means that

m„0 e A„0 n U*Q c A„0 + 1nV*0 + 1.

Taking into account the Jf-convexity of the set U, we have that:

x = ax + (a 2- a 1) + . . . + ( a „ 0- a „ 0_ 1) + u„0e (F 1* П Л Л +

+ (T2* n A2) + ... +(V„* n AnQ) + (V*0 + 1 n A„0 + l ) c= U.

Hence, indeed, F c U . ■

Theorem 2.2. Л sequence (x„)*=1 о/ points of E is po/-convergent to x if and only if the following conditions hold:

(i^ There exists an index j 0 such that x„ eAj 0, n = 1,2,...

(i2) (xn)n=i ™ p-convergent to x.

P r o o f o f necessity. Let us notice that, by Theorem 1.8, we can require the elements of sé to be /х-closed. Moreover, we can suppose that x = 0.

It follows directly from Theorem 1.2 that condition (i2) holds. Let us assume that condition (i^ does not hold. Since is an increasing family, then there exists a subsequence (xnj)jL x such that хп.фд] ; j = 1 ,2 ,3 ,... Sets Aj\ j = 1,2, 3,..., are /х-closed and therefore we can find a neighbourhood U* e M (p) such that хп.фА] + и * .

00

Hence xn.ф f) ( A j + U f ); j = 1, 2, 3,...

j = 1

But this contradicts the pw -convergence to 9 of the subsequence (xnj).

P r o o f o f sufficiency. Let us now assume that a sequence (x„)®=1 satisfies conditions (ij) and (i2). Then, this sequence is p | ^-convergent and therefore, by Theorem 1.3, it is also p^ | -convergent.

In this way the theorem is proved. ■

Corollary 2.1. A sequence (x„)®=1 of points of E is p^-Cauchy sequence if and only if there hold condition (i^ of the preceding theorem and

(i2) a sequence (x„)®=1 is a p-Cauchy sequence.

Indeed, it is enough to notice that in an n.a. space a sequence (x„)®=1 is a /х-Cauchy sequence if and only if a sequence (xn + 1 — x„)®=i is convergent to 0.

T^heorem 2.3. Let us assume that pi is a metrizable topology, F — a locally Ж -convex space. A linear map f from E into F is p^-continuous if and only if it is sequentially p^-continuous.

Proof. Suppose that / is //^-sequentially continuous but it is not /Е/-continuous. In this case, by Theorem 1.3 (b), there is an index j 0 such that f\Ajo is not /x|AJo-continuous. Therefore, for some neighbourhood V o id in F, a set (/ | А]о)~ х (L ) = / ' 1( f ) n 4 j() is not a p | ^-neighbourhood of в. It follows from the assumption of the //-metrizability of E that there is a family = (U*)^=1 forming a p-base of neighbourhoods of в. Then

« = 1 ,2,3 ,...

Consequently, there must exist a sequence (x„)®=1 of points of AJq, p-con­

vergent to в and such that xn$ f ~ x{V) for n = 1,2,... It follows from Theorem 2.2 that this sequence is //^-convergent to в and f { x n)

= (f\AJo) {xn)$ V; n = 1, 2,...

Hence a sequence (/(*„))*= i is not convergent to 0. But this is in contradiction to the assumption of the sequential continuity of /.

Proof of necessity is easy. ■

Theorem 2.4. Let p be a metrizable topology. The topology p^ is the only locally Ж -convex topology satisfying the two conditions (h) and (j2) simultaneously:

(jj) A sequence (xn)®=i of points of E is т-convergent to x if and only if it is p-convergent to x and xne A j(); n = 1,2,... for some j 0.

(j2) A linear map f from E into a locally Ж -convex space F is т-continuous if and only if it is т-sequentially continuous.

P roof. We must show that if т satisfies conditions (ji)>(j2)> then

t = p^ . Let us consider the identity map i : (E, р^) ^( Е, т) .

It follows from Theorem 2.2 and (ji) that i is //^-sequentially continuous.

Therefore, by Theorem 2.2 it is p^ -continuous. In a similar manner we prove the т-continuity of a map i -1 (by (h), Theorem 2.2 and (j2)); ■

Theorem 2.5. A set B cz E is p^-bounded if and only if it is p-bounded and E cz AJo for some j 0.

P roof. Since p ^ P,?, so each set //^-bounded is //-bounded. Let us assume that В is contained in no set of a family Ж . Let (a„)*=1 be a sequence of points of Ж , convergent to 0. By (1Д we find an increasing sequence of integers (&„)*= i such that

c a« A kn + l ; n = 1,2,...

Next, we choose a sequence (x„)®=i of elements of the set В , satisfying the condition : xn ф Akn + ; n = 1 ,2,3 ,...

Then ос„хпфапАкп + 1, therefore оспх„фАкп for n = 1 ,2 ,3 ,... As a result, а„хпф Aj\ j ^ k„; n = 1, 2, 3,...

Hence, by Theorem 2.2, the sequence (a„x„)*=1 cannot be ^-convergent to 9. But it contradicts the assumption of the boundedness of the set B.

Hence В <= Aj for some j 0.

P r o o f o f sufficiency. Let (x„)®=1 be any sequence of points of the set В and (a„)”=1 a sequence of elements of Ж , convergent to 6. We may assume that |a„| ^ 1 for n = 1 ,2 ,3 ,... Then anxne A jo for some j 0 and the sequence (a„x„)“=1 is ^-convergent to 9. But, by Theorem 2.2, this means that a sequence (a„x„)*=1 is /^-convergent to 9. Since this holds for each sequence (x„)®=1 and (a„)®=1, so В is цы-bounded.

Now we consider the following situation:

We have an n.a. space (E , p) with the second locally Jf-convex topology v.

Let us assume that there exists a v-bounded neighbourhood of 9 in the topology v. It is equivalent to the fact that (E , v) is a normed space (see [9] and Theorem 1.1).

Let S be a fixed JT-convex and bounded neighbourhood of 9 in the topology v. Let us choose an arbitrary sequence (a„)*=1 of points of Ж, satisfying the conditions:

(c i) M < |a2l < •••

(c2) Hm |aj = QO.

00

Let us denote: A„ := anS; n = 1 ,2 ,3 ,!.. It is now easy to see that sé := { A„: n = 1 ,2 ,...} possesses all properties required in the preceding sections. Hence, by Theorem 2.1, a base of ^-neighbourhoods of 9 is composed of all sets of the form:

00

(B4) V = Fo* n П (H* + <XiS), where Ц* е &( ц ) for i = 1,2,...

i = 1 .

It follows directly from Corollary 1.1 that:

Theorem 2.6. I f (/?„)*=! is another sequence satisfying conditions (ct), (c2), and S' — another Ж -convex and bounded neighbourhood of 9 in the topology v and s/' := {finS': n = 1 ,2 ,3 ,...}, then

М/ = 9-r-

From the theorem it follows that the choice of a sequence (a„)®=1 and a set S is not important. Therefore, next we shall additionally assume that S is a v-closed set. We shall call the topology pr/ so defined a bitopology and denote it Ъу y(p, v).

Theorem 2.7. The following conditions take place:

(a) if p ^ v, then y(p,v) ^ v, (b) if v ^ p, then y(p, v) = p, (c) if v = y(p, v), then p = v.

P roof, (a) For an arbitrary y{p, v)-neighbourhood V of в of the form (B4) we have

00 00

v = V* n П (Vj* + « j S) => V0* n П otjS = V0* n ( a i S).

j = i

But since p ^ v, so F is a v-neighbourhood of 0.

(b) If v ^ p, then all elements of the family sé have non-empty /i-interior.

Therefore, by Theorem 1.6, we obtain y( p, v) = p.

(c) It is enough to show, by Theorem 1.2, that v ^ p. From the properties of the topology v it follows that the family

S? := {PS: О Ф f i eJf }

forms a base of v-neighbourhoods of в. For an arbitrary fixed scalar P Ф 0 we find an index j 0 such that \P\ < |aJo|. Then PS a <x.JoS. Since v = y(p, v), by Theorem 1.4, there are sets F*, F2*, V * , ... G^ ( p ) such that

00

F := Jf -conv (J ( Vf nc i j S) a ps.

j =1

We can assume that V} c= Vj + l ; j — 1 ,2,3 ,...

Finally we shall prove that Vj* c ps.

Let us suppose that there exists a point x e Vj*\ps. Then we find a scalar AeJT, |A| ^ 1 such that otxeccj()S\pS. Since Vj* is a balanced set, thus Ax g Vj* n ctjQ S cz F c pS. m

Theorem 2.8. Let D be an arbitrary v-bounded set. Then y{p, v)|D = p\D.

P roof. Since S is an absorbing set, s o D c aJQS for some j 0. It follows from Theorem 1.3 (a) that

y(p, v)| D = (y(p, v )I aJo S) I D = (p\aJoS)\D = p\D.

The symbol Bd(t) denotes a family of all т-bounded subset of E. It follows from Theorem 2.5 that:

Theorem 2.9. Bd(y{p, v)) = Bd(^) n Bd(v). I f p ^ v, then Bd(y(/x,v))

= Bd(v).

In Theorem 2.3 we proved that if p is a metrizable topology, then the ju,-sequential continuity and ^/-continuity of linear maps are equivalent.

It is a specific property for Ж -bornological spaces. In spite of the above, we shall show that:

Theorem 2.10. Assume that p ^ v and ( E, y( p, v) ) is а Ж -bornological space. Then p = v.

The proof follows from Theorems: 2.9, 2.7 (a), (c).

Corollary 2.2. I f p < v, then the topology y{p,v) is not metrizable.

Theorem 2.11. Let p ^ v, and let there exist a closed and v-bounded neighbourhood t/enbh(v). I f (£, у(p, v)) is a Ж -barreled space, then p = v.

P roof. We can assume that U is a Jf-convex set. Then U is а Ж -barrel in the topology y(p,v) and consequently, it is a y (p, v)-neighbourhood of в.

Since the family {a!/: 0 Ф a e J f'} forms a base of v-neighbourhoods of в, so v ^ y(p, v). Now, by using Theorem 2.7 (a), (c), the proof is completed. ■

Corollary 2.3. I f p < v and there exists a p-closed and v-bounded neighbourhood of в in the v-topology, then:

(a) (E, y ( p , v )) is not a Baire space.

(b) E is not a у (p, v)-reflexive space.

Now we shall study some properties of a space topologically conjugate to (E , y(p, v)).

Let E\,E'2,E'3 denote, respectively, a conjugate space of spaces (£, p), (E, y (p, v)) and (£ ,v). Let us assume that a field Ж is spherically complete (i.e. every set of spheres totally ordered by inclusion in the field Ж has a common point).

Theorem 2.12. A space E'2 is complete in the topology of the uniform 4 convergence on у (p, v)-bounded sets (i.e. /?(£ 2, (E, y (p, v)))-complete).

I f F ^ v, then fl (E 2, (E , y (p, v))^ — P ( £3, ( E, v)) | E2, and E 2 is a complete subspace of ( £ 3 , /?(£з, (E, v))) (where in the space £ 3 ( £ 3, (£, v)) is the topology of uniform convergence on v-bounded sets).

P roof. Let (f)teT be an arbitrary M - S sequence of points of E'2 such that it is a f ( E' 2, (£, y(p, v)))-Cauchy sequence. Then, for each x e £ , an M - S sequence (f ( x ) ) teT of points of J f is a Cauchy-sequence, and so, it is convergent to g(x).

The functional

g: Е э х - > д ( х )

is linear. It follows from Theorem 2.9 that an M - S - sequence (/|a„S)r6r of p\ a„S-continuous maps is uniformly convergent on each set anS and therefore g\oc„S is p I oc„ S-continuous. Since this holds for each index n, so g e E 2. m

Theorem 2.13. Let a field Ж be discretely valuated, and p < v. Then E2 is equal to a closure of the set E\ in (£ 3, /?(£'3, (£, v))).

Lemma 2.1. I f x is a locally Ж -convex topology in a vector space F , A — closed set, В — a c-compact subset of F, then A + B is a closed set.

Indeed, it must be shown that for any point хф A + B there is а Ж -convex neighbourhood U of в such that (x — U) n (A + B) = 0 . This is equivalent to the condition:

( x - B ) n ( A + U ) = 0 .

Assume that this equality is false. Then a family : Ж : = {(x — B) n n ( A + U): U g Jf-nbh (t)} is a base of a filter on x — B. It is easy to see that it is a base of а Ж -convex filter (i.e. such a filter that it possesses a base composed of sets of the form y + A, where y ex —В and A is Ж -convex subset of x — B). Indeed, for each U e JT-nbh(r), we have ( x - b ) n ( A + U ) = z + C, where z is an arbitrary point of the set (x — B)r\

n ( A + U) and C := ((x — B) n (A + U) — z')) is a Jf-convex set. It follows from the assumptions that x — B is a c-compact set. (i.e. every Ж -convex filter on x — B has a limit point ye x — B). And therefore a filter generated by the base Ж has a limit point ye x — B. On the other hand, it is evident that

ye A + U a A + U + U = A + U for each U e Jf-nbh(t).

Hence ye A = A but this contradicts the assumption that хфА + В. я To a proof of the theorem we shall need the following definitions and theorems (see [9]).

A subset A of a locally Ж -convex space E is said to be Г -closed, if for any point x0eE\A there is a continuous n.a. seminorm p satisfying conditions p (x0) > 1, and p (a) ^ 1 for a e A . A will denote a Г -closure of a set A,

Theorem A. Let E be a locally Ж -convex space. Then:

(a) Every Г -closed subset of E is Ж -convex and closed.

(b) I f Ж is a discretely valuated field, then every Ж -convex and closed set is T-closed.

(c) There exists a base of neighbourhoods of 6 composed of Г -closed sets.

(d) Г -closure of any bounded subset of E is a bounded set.

(e) Every Г -closed subset A cz E is Г -closed in topology сt(E,E' ).

Theorem B. Let E be a locally Ж -convex Hausdorff space over a spherically complete field Ж , A — subset of E. Then:

(a) I f A a E, then (A0)0 = A.

(b) A0 is a Г -closed set and (£', о (E ', E))-closed set.

(c) I f M is a family of о (E, Enclosed, Ж -convex subsets of E, then ( П B)° = ( (J B°) ,t<£ £)5 where ~e (E' ,E) denotes Г -closure in topology

Bert Bert

ч № ',£).

P r o o f of theorem . It follows from Theorem 2.12 that E'2 is

а p ( E3, ( E , v ))~closed subset of E3 and therefore it is sufficient to show that E\ is a f$(E'3, (E, v))-dense subset of E'2.

Let В be any v-bounded set. Then its Г -closure В is a Jf-convex and v-bounded set. (By Theorem A (a), (d).) It follows from B(b) that В is a Г -closed set in the weak topology a ( E , E 3), too, and hence it is ( j ( E , E 3)~closed.

Let f e E 2 and s > 0. Since pi^ \B = pi\B, there exists a Г -closed neighbourhood l/enbh(t) such that \f (x)\ ^ s; x e B n U . (By Theorem A(c).) As a result, / ee(B n U) ° . Since pi v, so U is a Г -closed set in the topology v. Repeating the argumentation analogous to the previous, we have that U is a a ( E , E '3)-closed set.

Let us notice that the JT-convex sets

B°

Hence

B°

a.

(B n U f =

(B°

(B°

B°

Consequently, / ee(B° + U°). Hence there exists a linear functional g e e U 0 such that f - g s

eB

At the same time, it is observed that g is bounded on a pi-neigh­

bourhood U of в, and therefore it is /i-continuous.

Since |/(x:) — #(x)| < £, x e B , so the free choice of В and e completes the proof. ■

Finally, we will consider some example of an n.a. bitopological space.

Example 2.1. Let Г be a non-archimedean FK-space over Ж, i.e.

a locally Jf-convex, metrizable and complete space of sequences, such that a topology Ц* of a space F is finer than a topology induced from s(Jf) to F (s p O is a space of all sequences of points with the usual topology).

Assume that F =э ( Ж) (с ( Ж) is a space of all convergent sequences of elements of Ж) . Denote by pi, v, respectively pi*\ с( Ж) and a topology of the space с ( Ж) generated by the norm ||(xi)? i1|| = sup|x,|. A pair (pi, v)

VEjf defines some bitopology у (pi, v) in с (Ж ).

Let us examine some properties of this topology in the case of pi ф v.

We observe that ( с( Ж) , v) is an n.a. F K - space and therefore pi < v is a simple consequence of the closed graph theorem.

A topology y(pi,v) can be considered as a mixed topology generated by the topology pi and the family of sets Ж = (a„5: n = 1 ,2 ,...}, where (an)*=i is а sequence of points of Ж , such that 1 = |ai| < |a2|

< ..., lim |a„| = oo and S := { х е с ( Ж ): ||x|| ^ 1}.

n-> 00

We shall demonstrate that all sets of the family Ж have an empty /r-interior.

Indeed, if this is not the case, then the set S would have a non-empty

д-interior and consequently, S would be д-open (see [5]). Hence v ^ ц which contradicts the assumption.

We shall now prove that all elements of family я/ are д-closed. Let us assume that this is false. Then S would not be д-closed and there would exists a sequence x2 = (/?2i),“ 1, ••• of points of c (Jf), д-con­

vergent to x0 = iPoi)ï°=i<tS-

Then, for some index i0,\Poi0\ > 1-

But it is false because F is an FK-space and therefore lim fini = j30, n~> 00

i = 1 ,2 ,3 ,...

It follows from the above consideration, by Corollary 2.2 and 2.3, that Д < y(F, v) < v.

The space ( c p f), у(д, v)) is neither metrizable nor reflexive. It is not a Baire space, either.

References

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[4 ] J. B. C o o p e r , T h e str ic t to p o lo g y and spaces with m ixed topologies, Proc. Amer. Math.

Soc. 30 (1971), p. 583-592.

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Wet. 62 (1959), p. 391^105.

[8 ] A. С. M. V a n R o o i j and W. H. S c h ik h o f, N on-archim edean analysis, Nieuw Archief V oor Wiskunde 19 (1971), p. 120-160.

[9 ] J. V a n T ie l, E spaces localem ent Ж -co n vex es ( I —III), Proc. Kon. Ned. Acad. Wet. 27 (1965), p. 248-289.

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INSTITUTE OF MATHEMATICS UNIVERSITY OF CODZ