Minios’ theorem in nonarchimedean locally convex spaces

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1990)

An d r z e j M ^d r e c k i

(Wroclaw)

Minios’ theorem in nonarchimedean locally convex spaces 1. Introduction, notations and preliminaries. Let G be an abelian group.

A complex-valued function / defined on G is called positive-definite if (1.1) /(0) = 1 and Z cJidi-g^Cj >0,

i.j= i for all integer n, c{eC and ^.

G.

If G is an abelian topological group then by PC{G} we denote the set of all positive-definite and continuous functions on G.

Now consider a triplet (X,

Ж) where X is an arbitrary set,

is a er-algebra of subsets of X and Ж is a compactology in (X,

(cf. e.g. [2], [17]), that is, a class of subsets of X such that

(C) for every sequence {Kn} (K n

Ж , n = 1, 2, ...) such that p)«+=°°i Kn = 0 , there exists a positive integer N such that /£„ = 0 .

We call every such triplet (X,

Ж) a Radon triplet. Every set function p such that

(Rl) p(X) = l,

(R2) p is a cr-additive measure on

(R3) p is Ж -tight ([1]), i.c., for each £ 0 there exists K{ 6 JT such that p(KE) > l - s ,

is called a Radon measure on (X,

Ж) (cf. [21]).

By 01 {(X,

, Ж)} we denote the set of all Radon measures on the Radon triplet {X,

, Ж).

If G is an abelian topological group then we write 0t{G} for

^{(G, 01(G), J f (G))}, where (G, 01(G), (G)) is the standard Radon triplet on G, i.e., J'(G) is the а-algebra of all Borel sets in G and Ж (G) is the family of all compact subgroups of G.

If G is an abelian topological group then by G we denote its Pontria- Qin dual, that is, G is the abelian topological group of all continuous *

* 1980 Mathematics Subject Classification: Primary 43A32, 43A35, 46P05; Secondary 60B11, 60B15.

102 A. Mqdrecki

characters on G (i.e., continuous homomorphisms from G into the unit circle S1 = {

C: \

\ = 1}, where C is the field of complex numbers), endowed with the topology of uniform convergence on precompact sets in G.

Each /ieâ${G} defines the characteristic functional (ch.f. for short) fi on G by the formula

(1.2) fi(g*) = Sg*{g)n(dg), g*eG.

G

Obviously, /lePC{G}. We set Ê{G) = {fi: peM(G)}.

Each geffl{G} defines the inverse characteristic functional (i.ch.f. for short) fi on G by the formula

(1.3) fi{g) = [g*(g)n{dg*), geG.

G

Obviously,

U

PC{G}. Moreover, we set .#{G} = {fi: ju<=M(G)}.

We say that an abelian topological group G is a Sazonov group if there exists a topology £~s on G such that

(SC) P C { ( G , ^ s)} = â{G}.

3TS is called a Sazonov topology on G (cf. e.g. [10], [16]).

We say that an- abelian topological group G is a Minlos group if there exists a topology ?TM on G such that

(MC) PC {G}=â{(G,3TM)}.

is called a Minlos topology on G.

Let ^ be a subcategory of the category TAG of all abelian topological groups and continuous homomorphisms. Let S(^) and M(^) denote the subcategory of all Sazonov groups and all Minlos groups in respectively.

The Bochner problem for consists in describing S { and M(^) for given

^ ç TAG. More precisely, let Kat(TAG) be the category of all subcategories of TAG. Of course, S and M are functors from Kat(TAG) to Kat(TAG). Then the Bochner problem consists in calculating S(^) and Mifê) for the most interesting points (objects) ^ of Kat(TAG).

In the following important cases the form of Sfi?) or M(^) is well known.

(1) Let # = LCA be the category of all locally compact abelian groups.

By Weil-Raikov’s theorem (cf. e.g. [6]) and Pontriagin’s theorem (G = G), S(LCA) = M(LCA) = LCA.

(2) Let # = Hilb(R) be the category of all real Hilbert spaces. Then M(Hilb(jR)) = (finite-dimensional spaces over R} and S(Hilb(/?)) = Hilb(R) (Gross-Sazonov’s theorem, [3], [5], [21]).

(3) Let <€ = Ban(/?) be the category of all real Banach spaces. Then М(Вап(Я)) = M(Hilb(R)) and 5(Вап(Д)) $ (BeBan(R): В is of cotype 2}

(cf. [7], [15], [22]).

(4) Let <€ — LCWm{R) be the category of all metric locally convex spaces

over R. Then by Minlos’ theorem ([12], [13], cf. also [2], [22]) M(LCVm(R)) is

the category of all nuclear spaces over R.

Milnos' theorem in nonarchimedean locally convex spaces 103

(5) Let '(> = Bans(F) be the category of all separable p-adic Banach spaces over a fixed local field F. Then S(Bans(F)) = Bans(F) [10, Theorem 4.3].

In this paper we give a solution of a half of the Bochner problem (the Minlos problem) for the subcategory LCV(F) of TAG, of all Hausdorff locally convex spaces over a local field F . Namely, we prove that M(LCV(F)) = LCV(F), for an arbitrary local field F.

The form of S(LCV(F)) is more complicated. In particular, S(LCV(F)) ^ LCV(F) (cf. [11]), but we will not discuss it in this paper.

2. Sazonov and Minlos spaces over local fields. Let F be a local field (cf. [10]). An F-linear topological space V is called a nonarchimedean locally convex space over F (locally convex space for short) if its F-linear topology 3~v is generated by a family IIv — {pa:

Iv) of nonarchimedean seminorms on V (cf. e.g. [18, p. 62]), that is, 3TV is the weakest topology on V under which all functions from Пу are continuous.

A further discussion of nonarchimedean locally convex spaces can be found in [14] and [18].

Let F be a locally convex space over a local field F (l.c.s. for short), and V*

the set of all F-linear continuous functionals on V. We denote by <•, •) the duality function: Vx V*

(

, г>*)ь-*<(и,

* }

F. By ^(F*) we denote the algebra of all cylinder sets in V*, that is, the family of all sets C of the form (2.1) C = C(Vl, ..., vn; В) = {г* e V*: ((v,, F*>, ..., (vn,

*))

B], where vt

V and В is a Borel set in F".

Let W be a finite-dimensional subspace of V. Then by (é>(V*, Щ we denote the family of all cylinders C of the form (2.1) with B ^ W. It is trivial that

W) is a a-algebra for every finite-dimensional W.

We recall that a cylindrical measure v on V* is each finitely-additive set function on tf(V*) such that v(V) = 1 and v is rr-additive on ^(V*, W), for every finite-dimensional W ^ V .

The following lemma gives the relations between the cylindrical mea­

sures on V* and Radon measures on the standard Radon triplet (F*, ЩУ*), X(V*%

L

2.1. There exists a one-to-one correspondence between (1) the elements

&{V*}, and

(2) the cylindrical measures v on #(F*) such that

(2.2) far every s > 0 there exists KsEJf{V*) such that for each C

^(V*) with C o K £ = 0 we have v(C) < e.

Lemma 2T is an easy modification of Prohorov’s theorem [22, VI.3,

Theorem 3], where F = F-lin{<i>, •):

V}. By Lemma 2.1 there exists

one-to-one correspondence between the Radon measures ve^{F *} and

104 A. M^drecki

the i.ch.f. vePC{F}, which in this case has the form (2.3) v(u) = f %F{(v, v*))v(dv*), veV,

y *

where X

is the standard additive character on F (cf. [10]).

Let F be a local field.

De f in it io n

2.1. A l.c.s. V over F is called a Sazonov space if there exists a linear topology on V* such that

(2-4) PC{(V*,3TS)} = â { V j .

is called a Sazonov topology on V* (cf. [10]).

De f in it io n

2.2. We say that a l.c.s. V over F is a Minlos space if there exists a linear topology $~M on V* such that

(2.5) PC {V} = É{(V*,3rM)}.

2TM is called a Minlos topology on V*.

3. Two important examples of l.c.s. associated with p-adic Banach spaces.

Let (В, || • ||B) be a separable p-adic Banach space over a local field F (cf. [14], [18, p. 4]).

The space B + 00

De f in it io n

3.1 (cf.

An F-linear continuous map S

B*->B is called nuclear in В if lim„ ||S(en)||B = 0 for some orthonormal base {en} in В (cf. [14]).

We denote by N F(B*, В) the class of all nuclear maps in B. For each S e N F(B*, В) we define a pseudonorm ps on B* by

(3.1) ps(x) = ||S(x)||B, xeB*.

By nF(B*, B) (cf. [10, Sect. 3]) we denote the weakest topology on B*

under which all seminorms ps are continuous.

By B + o0 we denote the F-linear space В* endowed with the nuclear (normal) topology nF(B*, В), that is, B + o0 — (B*, nF(B*, В)). It is easy to see that B +00 is a Hausdorff l.c.s. over F.

Le m m a

3.1. Let В be a separable p-adic Banach space. Then B + o0 is a Polish space, i.e., a separable complete metric space.

P ro o f. B +00 is metric (cf. e.g. [20, p. 37]). B +o0 is complete. Assume that b = {bn} is a (countable) Cauchy sequence in B +oa, that is, for every S e N F(B*, В) the sequence {S(bn)} is a Cauchy sequence in B, and therefore f b(S) = UmnS(bn)e B exists. It is easy to verify that the mapf b: N F(B*, B)-+B is continuous, since f b = limn/ n is the pointwise limit of the F-linear continuous maps /„: N F(B*,B)^>B defined as f n(S) = S(bn), n ^ l (observe that the F-linear space N F(B*, B) is a separable p-adic Banach space with the nuclear norm

(N) ||5||N = sup||S(e„)||B, S e N F(B*, B),

Milnos’ theorem in nonarchimedean locally convex spaces 105

where {en} is a fixed orthonormal base for В). Now,

\ \ № \ \

= ^{Щ Ъ М в} < ^Ш • \\Ь„\\в* < IIS IU - IlfeJlB*

and N

(B*, B) is complete (by a Banach-Steinhaus type theorem, cf. e.g. [18, Th. 1, p. 84]). Now, let {£„} be an orthonormal base in the separable p-adic Banach space N F(B*, В), i.e., for each S e N F(B*, В)

(3.2) S ^{< S ,£ „> }£ c0(f), || £ J = 1, ||S||N = sup|<S,Ê„>|F.

n

Since f b: N F(B*, B)-+B is F-linear and continuous, by (3.2) we obtain

+ 00 +00

(3.3) f b(S) = I <S, E„> •/„(£„) = <S, S E„ ■/„(£„)> = S(B),

n= 1 И=1

where В = En •/b(En) e B* corresponds to {fb(En)} e /°°(F) under the isomet­

ry (p*: 1 co(F) = B*. Thus we have shown the existence of BeB* such that f b(S) = limnS(bn) = S(B), for every S e N F(B*,B). Obviously, B= \im nbn in

B+00, which shows the completeness of B +00.

B + 00 is separable. Let Sp(F) be a separable set in F. Then the set Sp(B+00) = {b e B : <b, en) = 0 for all but finitely many n and <b, e„>eSp(F)}

(here {en} is some orthonormal base in В) is a countable dense set in B +cc (cf.

also [10, Proposition 3.1 (3)]). If В = c0(F) then we write c + 00(F) instead of c0(F) + 00. In the notations of [10], c + 00(F) = (/°°(F), nF(/°°, c0)).

The space В _да. Let {e„} be an orthonormal base in B. For each X = {A„}

such that

(3.4) lim \Xn\F = +

2„eF\{0} for л > 1, П

we define an F-linear subspace Bx of В by

Bx = {xeB: there exists 0 ^ M(x) <

such that

|<x, en}\F ^ M(x)|A„-1 |F for n ^ 1}.

Let us equip the space Bx with the p-adic norm:

Px(x) = sup \X„• <x, en)\F (< M(x)), x e B x.

n

It is easy to verify that for each À = {A„} which satisfies (3.4), Ek — {Bx, p j is an infinite-dimensional p-adic Banach space over F. Let A be the set of all к = {Дп} which satisfy (3.4). Thus, we have a family {Ex. XeA) of p-adic Banach spaces, a linear space В and linear maps ix: Ex-+B (the canonical imbeddings) such that the family {ix{Ex): XeA} spans the entire space B. We define

В ж — limind Ek

ЛеЛ

(cf. e.g. [8], p. 74). That is, B-*, : = (B, where denotes the inductive topology. It is easy to see that B -^ is a Hausdorff l.c.s. over F. It is also easy to prove the following two lemmas:

L

3.1. (

_„)* = B +00.

106 A. Mtjdrecki

Le m m a

3.2. For every separable p-adic Banach space В ч п в +аа) = т в +ж).

R em ark 3.1. The Hausdorff l.c.s. cp(F), p = +

0,

(where c_ 00(F) = (c JF )),^ ) may be regarded as a nonarchimedean analogue of lp(R) spaces

p

4. Minlos spaces —positive results. Recall that if V is a metric locally convex nuclear space over R then we have the following theorem:

PC{F} =&{(V*,(TR{V*, V))}

due to Minlos (cf. e.g. [22, VI. 4, Theorem 4.3] or [2,111.1, Theorem 1.3]).

It is well known [19] that each l.c.s. over a local field F is a nuclear space in the sense of Grothendieck [4]. Therefore we can formulate the following

Th e o r e m

4.1 (p-adic version of Minlos’ theorem). Let V be a Hausdorff l.c.s. over a local field F . Then

PC{K} = .$t{(T*,<xF(F*, V))}.

To show that PC {V\ <= <rf (K*, V))} we need three lemmas.

Throughout this section, F will be a local field, | • |f its nonarchimedean discrete valuation and RF = {

F: \x\F ^ 1} the maximal compact subring of F.

«У || 'll O O we denote the “sup-norm” in Fn (n ^ 2). If p is a nonarchimedean seminorm on Fn then we write K p(r) for the r-ball { x e F n: p ( x ) ^ r } .

Le m m a

4.1. Let A be an arbitrary F-linear map from Fn into Fn. Set pA(x) = ЦТхЦ^,

F", and let HA be the probability Haar measure such that (cf. [10, Lemma 4.1])

ljeP4(i)(x) = I Z

« x ’ y » d H A(y), x e F n.

F r,

(Here and in the sequel l x(x) denotes the indicator of X .) Then ' f \\x\\KdHA(x) ^ ||T||N,

Fn

where || • ||N is the nuclear norm of A (see (N)).

P ro o f. Let M A be an Rf -submodule in F" such that M A = suppHA.

Obviously, M A is a compact convex absorbing set in F" (cf. [18, §2.5]) and O

M a. Therefore [14, p. 81], there exists a base el5 ..., en and positive real numbers a ls ..., anER+ such that

M A = {:x e F ": * = £ л ^ {, |A£|F ^ at}.

i= 1

Milnos’ theorem in nonarchimedean locally convex spaces 1 0 7

Since, for each i = 1, 2, n,

J XpiK^i, y » d H A(y) = j x A ^ d H ^ t ) =

KpJ1 )(^£)

F" F

~ ^ P l \ e i y R F ^ ’

H

is the probability Haar measure on pA{e^-RF (cf. e.g. [10, Lemma 4.1]; here HA denotes the distribution of the nonarchimedean r.v. <e(., •): (M A, HA)\-+

(F, &(F))). Hence at = рА(е^ that is, П

(4.1) M A = П PA(ei)' R

-

i = 1 From (4.1) we obtain

J 11*11 * dHA(x)= f sup |<x, «?,.>!FdHA( x ) ^ sup pA(et)

Fn Ma t 1

< SUp M^flloc = MlljV,

1 which completes the proof.

L

4.2. Let p be a probability measure in (Fn, ,#(F")), ft its ch.f. and A, В arbitrary F-linear maps from Fn into Fn. I f

|1 — Re/î(x)| ^ £ for all x e F n with pA{x) — ^ 1, then

(4.2) p({xeFn: pB(x) > p}) ^ e + (2/p)\\AB\\N (J8 > 0), where pB(x) =

x e F n.

P ro o f. Without loss of generality we can assume /? = 1.

Let HB be a probability Haar measure such that U PB(i)(x) = J

y » H B{dy), x e P .

F n

Since 1 — 1крв(1)(х) = 1 if PB(x) > 1, we have

(4.3) p{x: pB(x) > 1} = f p{dx) = j [ l - l KpB(i)(xy\p{dx) {PB>1}

= Re J HB{dy) J [1 - X

« x ’ y»]Mdx)

f n f n

= f V-Refi(y)lH.(dy).

F n

Under our assumptions

(a) $ [1- R e p(yJ]HB( d y ) ^ e } HB{dy)^e, I

^ 1}

(b) f [1 —Re/i(y)]Hs (dy) ^ 2 J HB{dy) 2 f pA(y)HB(dy).

[ PA>1} [ PA> 1} F n

(4.4)

108 A. M^drecki

But by Lemma 4.1

(4.5) J PÂy)HB(dy) = f \IAy\\„HB(dy) = f \\y\\„HAB(dy) a И В ||„.

p n p n p n

Combining (4.3), (4.4)(a), (b) and (4.5) we obtain (4.2).

No t a t i o n s.

Let F be a l.c.s. over a local field F . Let S ç F be an arbitrary subset in F. Then the polar set 5° of S is defined as follows:

S° = (u*

F*: |<y, v*y\F ^ 1, for each veS}

(cf. [14, p. 68]).

If p is a nonarchimedean seminorm on F then we write N(p) = {ve V: p(v) = 0} and we denote by Vp the p-adic Banach space which is the completion of the normed space (V/N{p), p), p(v + N{p)) = p{v), veV.

Le m m a

4.3. Let V be a finite-dimensional F-linear space and p a nonar­

chimedean seminorm on V. Let I p be the identity map in Vp. Finally, let p be a probability Borel measure in V* such that

|1—Re/2(x)| ^ £ if p{x) < 1 (e < 1).

Then for every j3 > 0

(4.6) ;<{K»VK?(/?)}«e + 2/!||/|)||„.

The proof of Lemma 4.3 is an easy modification of Proposition 1.2 from [2,111.1, p. 124] “modulo” our Lemma 4.2, that is, in the proof of Proposi­

tion 1.2 from [2] it suffices to replace Proposition 1.1 from [2] by our Lemma 4.2 and <т1{фм) by \\Ip\\N.

P r o o f of T h e o re m 4.1. ç . Let ZTV be the locally convex topology of F and let Ylv = (pf: i e l v} be a family of nonarchimedean seminorms on F generating

with pfV)

\F\P, for every

(Observe that if F is a local field then &~v can always be generated by such a family of seminorms [14,111.3.2, p. 33].) L e t/e PC {F} = PC{(F, f v)} be arbitrary. Then for every

£ > 0 there exists a finite subset F (fi) ç I v such that

(4.7) 11 — Re / (x)| < £ if pE(x) : = max pfx) < 1, xeV.

ieF(e)

Let K Pc(f) = {veV: pe{v) ^ /?}, e, /? > 0. Obviously, K Pe(f) is a compact set in the weak-star topology o>(F*, F) on V* (the Banach-Alaoglu theorem).

Let C(vt , ..., vn; B)

^(V*) (cf. (2.1)) be such that (4.8) C{ih , . . . , v n;B)nK°PeW) = 0.

Set L = F-linfvj, v2, ..., vn} and C(vl , B) = C(L; B). Without loss of generality we can assume that Be@(L*), C(L; B) = {v* e F*: ProjL*’i;*)GB}

and B = C(L; B) n L* (here ProjL* is the algebraic projection of V* into L*).

Milnos theorem in nonarchimedean locally convex spaces 109

By (4.8), C(L; В) £ (j8). Hence

B = C(L; B )nL * £ L* п{У*\К°рЩ ,

<4'9) L*\K%(P)nL° = Ь*\(КРе(Р) n L)°

(L* = L°), and K Pe(P) n L = {

L: pE(t;) ^ /?}. Now, let pi be the cylindrical measure on ÿ>(V*) which corresponds to / ( / = /r) and let Ie be the identity map in VPe. By the definition of pi, (4.9) and Lemma 4.3 we have

(4.10) M(C(L; B)) = nL(B) pL [L* \{Kp,(li) n L)0}

^ e + 2-jg||/e|L||JV ^ e + 2-[}• Ц/glljv = s + 2ft, where Ie\L denotes the restriction of Ie to L and piL is the сг-additive restriction of pi to <£(¥*, L). If we put p = e/2 then (4.8) and (4.10) imply that pi(C) ^ 2e for each C e^(F * ) such that С n K Pe(£/2) = 0 . But K 2e: = K°Pe(£/2) is a compact set in the oF(V*, F)-topology on F*. Hence by Lemma 2.1 and the definition of a Radon measure, pie&{(V*, o>(F*, F))} (in fact, pi admits an extension

(7F(V*, F))}) and / = fi, which completes the proof.

3 . Let pie3${(V*,

(V*, F))}. In order to show the continuity of fi on F it suffices to show the continuity of Re/ï at zero (cf. [10, Proof of Theorem 4.2]). By the definition of a Radon measure, for given e > 0 one can find a nonarchimedean seminorm pe on F which is «^-continuous on F and such that

l - e / 2 (r„>0).

Hence (cf. [10, Proof of Theorem 4.2])

|1 —Refi(v)\ ^ £ + 2 j |<u*, v)\Fdpi{v*).

But < (r‘}

f Kv*, u>|pdp{v*) sup |<i>*, v}\p = :qe(v)

K p E(re) v*e Kpe(r£)

and qe is «^-continuous. Hence |1— Re/ï(i;)| ^ £ + qe(v) if ve V, that is, Re/r is

^-continuous at zero (cf. [10, p. 234]).

An immediate consequence of Theorem 4.1 is the following generalization of the main result of [10].

Th e o r e m

4.2. Let В be a p-adic Banach space over a local field F. Then nF(B*, В) is a Sazonov topology on B*, that is,

PC {(B*, nF(B*, B))} = M{B]

or equivalently, PC{B+a0} = ${B}.

P roof. Without loss of generality we can assume that В = c0(I), for some set I = 1(B) (cf. [14, p. 44]). Since В is dense in B* in the topology nF(B*, В) (cf. e.g. [10, Proposition 3.1 (3)]),

PC {(B*, nF(B*, B))} = PC {(B, nF(B*, B))}.

п о A. Mitdrecki

By the p-adic version of Minlos’ theorem

PC {(B, nF{B*,B))}=â{B*,},

where B*n := (#*, (J

(B*, (J3, nF(B*, B)))). But each peM{B*„} is a er-additive measure on B* which is tight in the following sense (cf. proof of Theorem 4.1, В = c0(/)):

(4.11) for every e > 0 there exists /(г)ес0(7) such that

p

{({

0(/): ||Я(е)-х||С0(Л < 1})°} ^ l- 2 e . It is easy to see that (with /1(e) = ( / f(e)}, 1йпгЯг(е) = 0)

({xec0(I): ||A(e)x||co(/) ^ 1})°

= {{^i} ^/°°(/): |x,lF ^ l^(e)|F, for all iel} = :K £ °.

Hence by (4.11) and the above we find that: (1) suppp £ B, and (2) is a compact set in В (cf. [10, Proposition 4.3]). Therefore g.ed#{B}.

R em ark 4.1. Observe that Sazonov’s theorem for Hilbert spaces does not follow from Minlos’ theorem for nuclear spaces over R (cf. [22, VI.4. Exercise 2]).

The topology nF(B*, B) is especially interesting from the point of view of Bochner’s theorem in nonarchimedean locally convex spaces:

Pr o p o s it io n 4 .1 .

Let В be a separable p-adic Banach space over a local field F. Then nF(B*, В) is a Minlos topology on B, that is,

p c { B . œ} = â { ( B * , n F(B*,B))} = â { B +to}.

P roof. By Theorem 4.1

PC {B-oo} = ${(B + oo, cF(B + 00, B -J )} .

Each g e $ { ( B + œ, oF(B + X), В -«>))} is a u-additive (and # (B f ^-cylindrical) measure, which is tight in the following sense:

(4.12) for every г > 0, there exists an F-linear continuous map S : F; (for some X) such that

g { ( { x e B - œ. ||З Д ||В< 1 } )° } > 1 —2б. .

It is easy to verify that for each F-linear continuous map S : Ex-> B(X

Л), there exists 5 _1 e N F(B*, В) such that ({ x e B - ^ : ||S(x)||B ^ 1})° = {xeB + 0C:

l|S-1 Cx)||B < 1} =:7Cs. Since the sets of the form K°, S: Ех->В(ХеЛ), form a base of neighbourhoods of zero for B + o0 and since every compact set К in B + a0 is contained in some neighbourhood of zero in B + x , it follows that

+ ,#(B + J , Jf'(B + „))} £ .«{(B+oç, oe(B+„, B_,.))}.

But B + ao is a Polish space (cf. Lemma 3.1). Therefore each c-additive Borel

Milnos’ theorem in nonarchimedean locally convex spaces 111

measure is tight (cf. [1, Theorem 1.4], that is,

= &{(B*,ne(B*,B))},- which completes the proof.

Theorem 4.1 can be formulated in the following form:

Th e o r e m

4.l.A. Each Hausdorff locally convex space V over a local field F is a Minlos space and the topology oF(V*, V) (weak-star topology on V*) is a suitable Minlos topology on V*.

References

[1] P. B illin g s le y , Convergence of Probability Measures, Wiley, New York 1968.

[2] J. L. D a le c k ij and S. W. F o m in , Measures and Differential Equations in Infinite-dimensional Spaces, Nauka, Moscow 1983 (in Russian).

[3] L. G r o s s, Harmonic analysis on Hilbert space, Mem. Amer. Math. Soc. 46 (1963).

[4] A. G r o t h en d ie c k , Produits tensoriels topologiques et espaces nucléaires, ibid. 16 (1955).

[5] K u o H u i-H s iu n g , Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin 1975.

[6] E. H e w it t and K. R o ss, Abstract Harmonic Analysis, Berlin 1963.

[7] V. M a n d r e k a r , Characterization of Banach Space through Validity o f Bochner Theorem, in:

Vector space measures and applications I, Lecture Notes in Math. 644, Springer, Berlin 1978.

[8] K. M a u rin , Methods o f Hilbert Spaces, Monograf. Mat. 45, PWN, Warszawa 1972.

[9] A. M ^ d reck i, On Gaussian type measures in p-adic Banach spaces, preprint, 1983.

[10] —, On Sazonov type typology in p-adic Banach space, Math. Z. 188 (1985), 225-236.

[11] —, On the existence o f Sazonov topology in p-adic Fréchet space, preprint, 1986.

[12] R. A. M in lo s , Generalized stochastic processes and their extension to the measure, Trudy Moskov. Mat. Obshch. 8 (1959), 497-518 (in Russian).

[13] —, The extension of the generalized stochastic process to the complete additive measure, Dokl.

Akad. Nauk. Ukrainian SSR 119 (3) (1958), 439-442 (in Russian).

[14] A. F. M o n n a , Analyse non-archimédienne, Springer, Berlin 1970.

[15] D. H. M u s c h ta r i, Some general questions o f the theory o f probability measures in linear spaces, Teor. Veroyatnost. i Primenen. 18 (1973), 66-77 (in Russian).

[16] —, Sur l’existence d ’une topologie du type Sazonov sur un espace de Banach, Séminaire Maurey-Schwartz, 1975-76, No. 17.

[17] —, Sufficient Bochner topologies, in: Constructive Theory of Functions and Functional Analysis, III, Tzdat. Kazanskovo Universiteta, 1981, 64-72 (in Russian).

[18] L. N a r ic i, E. B e c k e n s t e in and G. B a c h m a n , Functional Analysis and Valuation Theory, Pure and Appl. Math. Dekker, New York 1971.

[19] M. van d er P u t and J. v a n T ie l, Espaces nucléaires non Archimédiens, Nederl. Akad.

Wetensch. Indag. Math. Ser. A. 5 (70) (1967).

[20] W. R u d in , Functional Analysis, McGraw-Hill, 1973.

[21] V. V. S a z o n o v , On characteristic functionals, Teor. Veroyatnost. i Primenen. 3 (1958), 201-205 (in Russian).

[22] N. N. V a k h a n ia , W. J. T a r ie la d z e and S. A. C ho b a n ia n , Probability Distributions in Banach Spaces, Nauka, Moscow 1985 (in Russian).

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