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

Séria I: PRACE MATEMATYCZNE XXX (1990)

**A****n 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 ^.*

e### 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, *

*s é ,*

*Ж) where X is an arbitrary set, *

*s é*

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

*s é )*

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

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

g* 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, *

*s é ,*

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

*(Rl) p(X) = l,*

*(R2) p is a cr-additive measure on *

^{s é ,}*(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, *

*s é ,*

*Ж) (cf. [21]).*

*By 01 {(X, *

*s é*

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

*s é*

*, 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 = {**

**S1 = {**

**z g****C: \**

**C: \**

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

**\**

**C**

*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,

j### U

g*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: *

*x e*

*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**

*b*

*(*

*v*

*, г>*)ь-*<(и, *

*v*

** }*

*e*

*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, *

*v*

**))*

*e*

*B],* *where vt *

*e*

*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**

**emma**

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

*v e*

*&{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*

*e*

*^(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 e*

*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*

*f*

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

*Let F be a local field.*

**D****e 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]).*

**D****e 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*

**D****e f in it io n**

### 3.1 (cf.

**[ 9 ] ) .**

*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.*

**L****e 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,*

**\ \ № \ \**

**\ \ № \ \**

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

^{Щ Ъ М в}

^{Ш}**\\Ь„\\в* <**

*and N*

*f*

*(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 = + *

**oo,**

### 2„eF\{0} for л > 1, *П*

*we define an F-linear subspace Bx of В by*

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

**+ o o**

### 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**

**emma**

** 3.1. (**

**В****_„)* = B +00.**

**106** **A. Mtjdrecki**

**L****e 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 = + *

**oo,**

### 0,

**—oc**

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

**(1 ^**

*p *

**< + oo).**

*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

**T****h 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 = {*

*x e*

*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 } .*

**L****e m m a**

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

*x e*

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

*ljeP4(i)(x) = I Z*

*f*

*« 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*

*e*

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

**..., anER+**

*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 ) = *

*l*

*KpJ1 * )(^£)

*F" * *F*

*~ ^ P l \ e i y R F ^ ’*

*H*

\4* 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*

*f*

*-*

*i = 1* From (4.1) we obtain

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

*Fn * *M**a** t* **1**

< SUp M^flloc = MlljV,

### 1 which completes the proof.

### L

emma* 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|lao, *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

^{X}

^{f}

^{K x > }*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}

^{F n}*= Re J HB{dy) J [1 - X*

*f*

*« 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*

*pa*

*^ 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).

**N****o 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* *

^{g}*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.*

**L****e 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

^{£ T V }*with pfV) *

**< =**

*\F\P, for every *

^{i g}**/ f**

**.**

### (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)*

*g*

*^(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 = {*

dg*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*, *

*gf*

*(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].

**T****h 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*

*f*

*(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

### {({

xgc### 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:

**P****r 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*

*e*

* 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:

**T****h 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).**

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**[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).**

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**[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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