On the theory oî sequences

R O C Z N I M P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

A. Go d z i e w s k i (Poznan)

1. In this paper we study the sequences in an arbitrary locally compact field К and we give the complete theory of sequence spaces.

It is known that every such field is a finite extension of one the following fields: the field of the real numbers, the field of the p-adic numbers or the field of the formal Laurent series on some finite field.

Moreover, the field К includes the minimal subfield Qr isomorphic to the field of the rational numbers, when the characteristic r = ⁰.

Kaplansky [3] proved that in an arbitrary local compact field there exists a non-negative real function T(-) such that:

(a) There exists a constant C > 0 such that T(t-\-s) < C[T(t) + T(s)], (b) T(t-s) = T(t)-T(s),

(c) T ( 0) = 0 and T(t) > 0 for t Ф 0.

In Sections 2-4 we prove, using the method of the metric groups theory, that on the subfield Q° this function takes the form T(-) = |*|1>, where 0 < p < oo, and it cannot appear in another form. Moreover, we prove, that the spaces l(pv), ⁰ < p v < ¹, are isometric to any subgroup of the metric group

where t,e K , 0 < qv ^ x < 1 for v = 1 , 2 , . . .

It follows from theorem of Pontriagin (see [10], p. 157) that the spaces l(pv) and the (*)-groups are identical with respect to isometry, if the field К is isomorphic to the field of the real numbers.

In Section 5 we consider a general properties of (*)-groups and in Section ⁶ the additive mappings on some sequences from this groups composed.

Sections 7,8 contains some remarks on a limit properties of (*)-groups.

2. Let R be an entire ring with a unit e and N (t), a non-negative real function defined on R.

On the theory oî sequences

oo

v = 1

292 A. G o d z i e w s k i

We put

» oo

Qn(x ) = У1 N{%) ^?

v = X

where x = (tv) is an arbitrary sequence in В and

OO

G6N = {X = (%)’• < °°}*

v = l

We shall give necessary and sufficient conditions in order that the set Qqn be a group. We prove the following lemma (see [6]).

2.1. Let M( t , s ) and N( t , s ) be non-negative real functions defined OO

on В x B . The convergence of the series M(tv, sv) implies the convergence

oo r**l

o f Z N (%, sv) if and only if there exist two constants C > 0 and e > 0 such

V = 1

that

N{t , s) < CM(t, s), when M ( t , s ) < e .

Proof. Since the sufficiency is obvious, we only prove the necessity.

We suppose that it is not true. Then there exist tn, sn such that N {tn1 sn)

~^nM(tn, s n) and M(tn, sn) < 1/n2 for n = 1 , 2 , . . . We may suppose that M(tn, sn) ф 0. We denote by hn the least positive integer such that

1 2

— < TcnM{tn, sn) < —- for n = 1, 2, ...

n2 ni

We write

. 1 . 1 ,/ , / ,/ , /

t\ — tx, sx — sx, t2 = j s2 = sx, .. . , , ski = Sj⁵

, r , f ___ , f ___ . f ___ .> ___ , ! __

tkx +1 = + 1 ~ %i + 2 = г2> Skx + 2 “ S2J •••? *k1 + k2 — ^2? Skx + k2 — S2?

^fc1+ft2+ ...+ * w_ i + l — S k x+ k 2 + ...+ k n _ x + 1 ~ ' Sn i •••>

/ , t

' ',k x+k2 + ...+ k n ~ s k x+k2 + ...+ k n s n •

Then

p o o o o o

y \ 1 ^

2 M ( t , K) = 2 M m , *„> < 2 < °°

J>=1 I>=1 V=1

and

OO . 0 0 o o o o

^ N (K, s') = J T kvN( t „ sv) ф Jj? v-TcvM{tv, sv) > = ^{0 0} >

V =1 ^{J> =} 1 ^{V = 1 V =} 1

a contradiction.

Co r o l l a r y. Putting instead of M(t, s) and N(t, s) — M(t) and N(t) oo

respectively we get by 2.1 that convergence of the series M(tv) implies

OO V = 1

convergence of ^ N (tv) iff there exist two constants C > ⁰, e > ⁰ such that

V = 1

N (t) < CM(t) for M ( t ) < e .

2.2. Th e o r e m. The following conditions are necessary and sufficient that the set GeN be a group :

(a) There exist constants C > 0, £ > 0 such that N(t-\-s) <

< C [ (t) + N (s)] for N (t) < e and N (s) < s,

(b) There exist constants D > 0, e > 0 such that N ( —t ) ^ D N ( t ) for N (t) < e,

(c) N (0) = ⁰, where ⁰ on left-hand side of this equation is the zero element of the abelian group of the ring B.

Proof. The sufficiency is trivial. To prove the necessity we suppose that Ge is a group and we denote M ( t , s ) = N(t) + B(s) and N ( t , s )

= N(t + s). Since ^qn (x) < oo and QN(y) < oo implies ^qn(x - \ -y) < oo,

OO OO 0 0

hence convergence of the series JC M(ty, s v) = N (tv) + 2 N(sv)

V = l V ~ l

OO 0 0

implies convergence of the series 2 N(t„,sv) — 2 N(tv + sv) and

applying ^{2 . 1} we obtain (a). v=l r=1

Since ^{qn (x)} < ^oo implies < °°i thus putting in 2.1, M(t, s)

= N (t) and N( t , s ) = N ( —t) for arbitrary t , s e В we obtain, that con-

OO o o

vergence of the series 2 M(tv, s v) = 2 N (tv) implies convergence of

OO OO V ~ 1 t — 1

2 N{tv, s v) = 2 B { —tv), hence (b).

I > = 1 V = 1

Condition (c) is obvious.

2.3. Let N (t) have the following properties:

(a) There exists a constant C > 0 such that N(t-\-s) ^ C[N(t) + N(s)]

for arbitrary t, se В ;

(b) N{t-s) =N( t ) - N( s ) for arbitrary t , s e B .

Then there exists к, ⁰ < x < ¹, such that the function N*(t) = [B (t)]x satisfies the inequality N*(e-\-t) < ¹ -\-B*(t).

Proof. By condition (a), we get B(t-\-s) < 20max[JV(t), JV(s)].

We take x such that (2C)* < 2. Then < 2 т а х [1 * (1 ), ^ ( s ) ] , hence

(1) -№*(<! + t2Jr . . . + t ) < 2k max B*(t{) for h = ¹ , 2, ...

Putting t{ = e for i = ¹ ,2 , ... , 2k we obtain

N*(2ke) < 2kN*{e) = 2k for Tc = 1 , 2 , ...

(2)

294 A. G o d z i e w s k i

We prove by induction that for an arbitrary positive number n the following inequality is true:

(3) N * ( n e ) ^2n.

Let 2k < n < 2k+l for ft = 1 , 2 , . . . Since inequality (3) is true for ft = 0, we prove (3) by induction with respect to ft. We have n = (n — 2k) -f-

+ 2*. If N*[(n — 2k)e] < N*(2ke), then from (2) it follows that N*{ne) = Wi‘[ ( w - 2 fc)e + ²fce ] <²W¹:(²&⁴ < ² -²fc< ²n.

If N*{{n — 2k)e']'^N*(2ke), then taking into consideration that n — 2k < 2k, applying the induction hypothesis we have N*(ne)

= N * [ { n ~ 2 k)e + 2ke] < 2N * [ ( n -2k)e] < 2 * 2 ( w - 2 ft) = à n ~2k+2 < 2n, whence we get (3).

Now we take an arbitrary t e R. Supposing that n is an arbitrary power of 2 we apply inequality (1) to the binomial decomposition (e + £)n-1.

By (3), we have

therefore _ .

jy*[(e + <)*_1] = V [ £ " “ Mi*

Lfc=0 ' ’

< wmaxJV*^7*'” 1

= %тах£.Ж*||№ 1| ej

whence

< 2%max^wfe 1j(^*(#))fc]

<2№[l + jy*(«)f“1,

+ < 2w[l +W*^)]"-1.

Since n may be as large power of 2 as we please, therefore N*{e + t) < 1 + JV*($).

3. It is known that every entire ring R may be embedded isomor- phically in a commutative field, namely, in the field of fractions of the entire ring R. We denote it R. Every element of R may be written in the form t/s, where te R, se 8 = J5\{0}.

Let J t = {Ж} be the family of all non-negative real homomorphisme on the multiplicative semigroup of the ring R. We set M[(p(t)~] = N (t), where q> is the isomorphism y: R -> R of the form t -+ts/s for te R, sc S.

Since <p{R) is the multiplicative subsemigroup with a unit e of the field R, thus J(* — {M} be the family of all non-negative real homo-

morphisms on y(R) <=■ R .

Let P ( • ) be a real function on В such that

and P ( • ) > M( - ) ^ 0, where M e Jt* and M is the maximal real homo­

morphism possessing this property. It follows from theorem of Kaufman (see [4], Theorem 3) that Jf(-) may he extended to a real homomorphism T(-) defined for all elements of the multiplicative group of the field B, such that P ( • ) > T( • ) > ⁰.

be a group, where T is one of thé possible extensions of Then T(-) has properties 2.2, (a), (b), (c) in the field B. Since T(-) is a ho­

momorphism, thus — -j = We denote by GtT the subgroup of the group GeT such that for arbitrary elements of В there exists a constant C > ⁰ such that

3 .2 . Theorem. (GtT*, gT*y , where T*( •) = [T( •)]*, is a metric Proof. By condition ( + ) and by multiplicative property of the function T(-), in view 2.3 we get that T*(-) satisfies the inequality

3 .1 . Let

(+ ) T

group.

whence

296 A. G r o d z i e w s k i

Moreover, T*( —t/s) =T*(t/s). Thus ет,(ж) = 2 T*(tjsv) has the V=1

following properties: ^{qt* (x)} = 0 o a ? = 0, where 0 = (0/£„); ^{qt, ( —x)}

= qt * (x), and ^qt *(x y) < Qt* {%) + Qt* (У)• The function g{x, y) = gT * (x — y) is an extended invariant matric, since ^qt*{x) may be infinite. Therefore we define the following subgroup of the group Gqt :

&L. = <x ~ ) : qt* (x) < oo

Obviously, GtT* is a metric group with well defined metric qt,(x, 0)

== Qt* («Г — 0 ).

3.3. Let

where 0 < pv < к < 1 for v = 1 , 2 , . . . We write

Qt*,pv(x)

Then (Gqt * (pv) , Qt*,Pv) is a metric group by 3.2 and 2.3.

4. Let К be an arbitrary field. It is known that the characteristic r of a field К is equal zero or a prime number l. There exists the minimal subfield Qr {r = 0 , 1 ) of a field K, where

{ /ïïb I

— e = (me)(ne)~l : m , n e B (the set of integers)!»,

n J

Ql = {0, e, 2e, . . . , ( l - l ) e } .

The set Q° is the field of fractions of the entire ring {me: me R}.

Let r = 0 ; then by 2 ,3 we have

(m \ m _ _

— 1 = — e is the isomorphism g : R ->Q° and R is the field n f m

of rational numbers. We set F = Tog, then

Ostrowski in [9] proved that for arbitrary archimedean norm w(-) in the field of rational numbers it is possible to find a real number s,

0 < s < ¹, that for all rational numbers mjn we have w(m/n) = \mfn\s.

Since F*( ’) is an archimedean norm on B, thus F*(m/n) = \m/n\s.

Hence

^ , m T I— e

n

m S/х m

n n , where p = sjx and ⁰ < p < oo.

Moreover,

OO

o o

V = 1

I® ’)-?*):

mv < oo

!•

< oo

!•

y i mv 2j I nv

v = l

< ⁰⁰ b where ⁰ < p v < % < ¹ and ⁰ < < ¹ for v = ¹ ,² ,

(riy. c Is • for ⁰ < s < ¹ ,

Thus where

■v= ¹

v i mv

^ nvv=l ¹ v is a metric on Gq *, and an JF-norm on the linear spaces Zs

G*T*(Pv) <= l(Pv) for ⁰ < ^ < я < Ü and v = ¹ ,² , . . .

It easily seen that the linear spaces Zs and l{pv) are completions of the metric groups (GeT, , £T*> and (GtT. (pv) , Qt*,pv)i respectively.

The Z(^„) spaces have been investigated by various authors (see e.g.

[2] and [7]).

5. Let К be a locally compact field. Then there exists (see intro­

duction) a non-negative real function T( - ) defined on К such that, by 3.2 and 3.3, the set

o o

СЛТ.(Р.) = {* = (O: Z < ос), where ⁰ < p v < к < I for v = ¹ ,² , . . . is a metric group.

5.1. <G*Tt{Pv), a complete metric group.

298 A. G r o d z i e w s k i

Proof. If xn = (tnv) is a Cauchy sequence in G*Tat{pv), then

o o

QT*,pv (Xn xm) (^nv ^mv)~\ v ~^ ^ r=l

for m , n - + oo, whence [T(tnv — tmv)Y v -> 0 for every v and m , n - > o o . Since the field К is complete, there exists a sequence (tv) such that [T(tnv — tv)]Pv ^-> 0 for every v and n - > oo.

«, However, [T(tv) f ^ [T(tv~ t nv) ] ^ + [ T ( t nv) f v for every v, thus

< ^oo, whence у = (t„) ^e GtT,(Pv). Moreover,

V = 1

00

QT*,Pv(®n- У ) = £ -> 0 for n -> oo,

*=i thus G*TAPv) is complete.

5.2. I f 0 < p v < qv < x < 1 for all v, then (a) GtT.(p v) £ GtT,(qv).

([3) The identity map <G*TAPv), Qt*,pv> -> <GtT.(qv), QT*,qf>

is continuous.

(Y) ^ T.{Pv) is ^nse in <GtTAlv), Qt*,qv>•

OO

Proof. We take ж = (<„)^e GtT„(pv), then JT [T(tv)]Pv < ^oo. Thus

» > = 1 o o

for large v, T(tv) ^ l and so [T(tv)Jlv < whence f£[T(tv)~\9v < ^oo

and xeGt T,(qv), thus (a). v=:1

Let xn = (tnv) € GtT,(Pv) and 9т*,р,(хп) 0 for n ^ oo. Then for w sufficiently large and an arbitrary e > 0

OO

V=1

thus T(tnv) < 1 for every v and so [T(tnv) f v < [T(ifM,)]14 We have

OO

У [T(inv)]3>’< г for n sufficiently large,

V = 1

hence QT*,gv(xn) 0 for n -> ^oo, thus ([3).

We denote by KeT„>Qv( 0, e) the ball of radius e > 0 and centre 0.

For x e KeT,sPv(0, min(l, e)) we have xeGtTt{Pv) and Xe K<4r*,qv№i e), thus in an arbitrary neighbourhood of 0 in the topology defined by gT* Qv we find an element belonging to G*T,(pv). Since the metric gT*,Qv is in­

variant, thus in an arbitrary neighbourhood of yeG*Tt{qv) we always find an element belonging to GeT,(pv), whence (y).

5.3. I f 0 < p v ^ q v ^ . x < l for all v and <GtT,(pv), дт*,р„У Щ se­

parable, then the following conditions are mutually equivalent:

(a) r(pv) is the topology induced on G*T*(pv) by r(qv), where r(pv) and r (qv) are the topologies defined by metrics Qt*,pv and QT*,qvi respectively.

(p) I f xn = (tn,)eGtT.(P,) and °°n ^{- * 0} r{qv), then xn ^ h in r{pv).

(ï) G*TÀPv) is closed in <GtT.{q,), qv>,

(S)

Proof, (a) =>.(y). If (a) is true, then qt*,Pv and QT*,qv give the same definition of Cauchy sequences in GtTt(pv), and so (G*Tt(pv), g T * , q v}

is complete, thus (y).

(ï) => ($)• It follows immediately from 5.2, (y).

(S) => (a). It follows from 5.2, ((3) and from the Banach Theorem on the inversion of homomorphisms (see [¹]).

(a) => ((3). Obvious.

((3) => (a). If ((3) is true, then the identity map <GtT„{pv) , т(д„)>

-> (G*TÀVv)i r (Pv)) is continuous at 0, hence is continuous everywhere (see [1]) and (a) follows from 5.2, (f3).

5.4. We suppose the 0 < r v, sv < к < ¹ for all v and we write p v

= min(rv, sv) and qv = max(rv, *,). Then GtT.{pv) = GtT,{rv) <^GtT,{sv) and G*Tt(qv) = G, where G is the smallest subgroup of the group G generated by G*T. (rv) и G*Tt (sv).

Proof. From 5.2, (a) follows that GtT.(pv) £ GtT,(rv) n GgTt(sv) and G £ GtT.{qv). Since

OO OO OO

S l T ( t , ) 7 ’ < max ( 2 [T(t.)7-, 2 [ T i o r ) ,

V = 1 V = 1 V = 1

thus GeT,{rv) n GtTt(sv) C GtT,{p,).

Let •

A = {v : rv > sv} and В = {v : rv < s„}.

If (tv)e GtTt(<lv), we write

К = tv for ve A and f' = ⁰ for ve В , t" = 0 for ve A and tf = tv for ve В .

Then OO OO

£ [ T ( t : ) Y - = 2 [T(t,)Y- < oo,

V^{= 1 ? = 1}

thus (t'v) e G*T,(rv) £ G.

Similary {t'f)e GtT,{sv) s Thus (t'v) |-(C) = (K)€ &- We have proved that GtTM ) S G, which gives GtT.(qv) = G.

5.5. The following conditions are mutually equivalent:

(a) GtT,{rp) £ GtTt(sv), (P) GtTt(pv) = G t Tt(rv), (r) ^^{2 , ( 0} = <#*.(&)•

300 A. G o d z i e w s k i

5.6. < 3 ^ (0 = 0 t T,{rv) iff GtTAp„) = в*тЛ&)‘

5.7. Theorem. I f 0 < rv, s„ < к < 1 for all v and <GtTt{pv), Qt*,Pv>

is separable, where p v and qv are defined in 5.4, then the following conditions are respectively equivalent :

(a) GÏT.(rv) Я GtT.(sv),

((3) r(rj induces a finer topology on GtT*{rv) nGtT*{sv) than r{sv), (y) if xn = (tm) e GtT* (rv) n GtT* (sv) and xn -> 0 in the topology x(r„), then xn 0 in the topology r(sv).

(«') GtT*{rv) = GtT*(sv),

([}') r(sv) and r(rv) induce the same topology on GtT*(sv) nGtT*(rv), (Ÿ) if xn = (tnv)€ GtT*(K)^GtT*(sv), then xn - +0 in the topology x(rv) iff xn~> 0 in the topology r(sv).

Proof, ^(a) => (P). If ^(a) is true, then, by 5.5, GtT*(pv) = GtT*(rv).

Hence and from 5.3, the topology induced on GeT*(pv) by r(rv) is identical with r(p„) and hence by 5.2, ((3) is finer than t($„).

((3) => (y). Obvious.

(y) => ^(a). We suppose that xne GtT*(pv) for n — 1 , 2 , . . . and xn -> 0 in the topology r{rv). We define sequences (yn) = (t'nv) and (zn) — (t'fy) as in 5.4. Then yneGtT*(pv), zneGtT*(pv) for n = 1 , 2 , . . . and yn -+ 0, and zn - + 0 in the topology x{rv).

However,

OO 0 0

У [2’(C )]'V = У [ ï ( O r

v=l v=l

and so zn -> 0 in the topology r(pv). From (y), yn 0 in the topology r(sv).

But

OO OO

У m o r = У [ 4 0 f ,

V = 1 K = 1

and so yn —> 0 in the topology t ( p y). Since (yn) + (zn) = (C) + (C) = Ю

= (xn), it follows that xn -* 0 in the topology r(pv). We have proved that if xn€ GtT*(py) for n = 1 , 2 , . . . and xn 0 in the topology r(r„), then xn -» 0 in the topology x(pv). Whence, by 5.3, GtT*(Pv) = GtT*{rv) and by 5.5, GtT*(rv) £ G*T*{sv). Thus (a).

(a'), ((У), (y') follow from (a), (p), (y), immediately.

6 . Let (p n v) be a sequence of real numbers such that p n+hv < p n>v for every v, 0 < p nv < к < 1 for n = 1 , 2 , . . . and lim p nv > 0 for every v.

П

Then, by 5.2, GtT*(pn>v) => G*T*{pn+l>v) and the identical injection of (GÏT.(p n+hv), QT*,pn+hv> into <GtT*{pn,v), QT*,Pn>t) is continuous.

A sequence 38 = {(GtT*{pn>v), 9т*,рп) ) is said to be pre-{G)-sequence.

We write (see [⁸]) (i) m = Gv(^>v),

oo

(U) nas = | n e ÿ w , 4 n=l

oo

where r denotes the topology of the group Q GtT. (pn,v) induced by metrics {QT*.Pnÿ ^ = ¹ , ² , ...), ^{n = 1}

OO

(iii) Qgg(x) = 2 Z~nQgg,n{x) for же| #| , where n—1

I eT*,pn>v(x)/(l + eT*,pn>v(®)) for XeGtT.(Pn,v),

Q&,n№) ' ’ ’ ^{„ .T V * , v}

l 1 for Же \38\ \GeT.(Pn,v),

(iY) P<% = {{GtT.{pn,v), QT*,pn>J: № =2> + l , P + 2, . . . } , M = the p-th element of the sequence 38.

q@(oc, 0) = q^(x — 0) is the invariant metric function on 38.

(v) [# ] = (Я, ея).

6.1. i c i 38 be a pre-{G)-sequence. We define for any r > 0 and natural n the following subsets of 38 :

K a {r) = {же \38\: дя ( х ) <г } and K®,n{r) = {же \n38\ : QT*tPnv(oc) < r}.

Then

(a) To every natural n and e > 0 there corresponds rj > 0 such that (b) To every e > ⁰ there corresponds a natural n and rj > 0 such that К я {е) ZD Kgg n(rj).

Proofs as in [⁸].

6.2. We take pre-(G)-sequences 38{ (i = 1, 2) and an additive mapping F of \38г\ into \38x\.

6.2.1. The mapping F is said to be nearly-open iff to every e > 0 there corresponds rj > ⁰ such that

с \Я1ж(кЯги)) => К щ М , where Cl^ denotes the operation of closure in [38 f\.

6.2.2. The mapping F is said to be open iff to every e > 0 there corres­

ponds rj > ⁰ such that

F( K^( e) ) = К Я1М .

6.2.3. The mapping F is said to be complete-closed iff the following condition holds: if (xn) ^e \38г\ satisfies Cauchy condition in [382\ and the

302 A. G r o d z i e w s k i

sequence (Fxn) tends in [&x] to some ye \&x\, then there exists xe \@2\ such that (xn) tends to x and у — Fx.

6.3. We tahe pre-(G)-sequences ^ (i = 1, 2) and an additive mapping F of the separable metric group \882\ into \BX\. I f to every p there corresponds kp such that F(\ p&2\) is of second Baire category in [kptfë-f], then T is nearly- open.

Proof. We can put all kp = 1 (see [8], Proposition 7). Let m be an arbitrary but fixed natural number. Por every xe \082\ there exists an open ball K x with centre x such that for x ', x" e K x we have q@2(x' — x")

< lfm. Since \&2\ is separable, thus we may replace the system of balls K x, xe \3§2\ by a countable subsystem K x, K 2, ... Then

o o i

W = U K n and x, ye K n => Q# ( x - y ) < — .

»= i 2 m

OO

Moreover, for every p we have \p&2\ c= U whence 1

OO o o

F(\p® 2 \ )<=J?( U Я.) = U F(Kn).

n=l П— 1

We denote Hn = F ( K n), then

OO

F( \ p&2\) ez (J Hn for every p.

7l~l

However, F(\p$tz\) is of second category in [^ J , thus there exists n0 such that HnQ is dense in any of these balls.

Hence there exist y 0e \У8Х\ and r > 0 such that

^ {У€ \B i \ : е ^ ( у - Уо ) < Л

thus there exists y xe H no such that д^1(у1 — Уо) < г1 = r l^ and C l # H no

=> {У* \Pi\- б я ^ У- Уо ) < r} => {y€ \SSX\: Q ^ i y - V i ) < « ? }•

From now on we repeat the proof given in [8]. Take ye \38x\ such that Q^( y) <r}. We have rj > Q^{y) = Q*1[{y1 + y ) - y 1'], whence у г +

+ У * С Ц Hno. Thus у е С Ц НПо - у х = СЦ (HnQ - у х) and (*) {У* \^i\- 9ях(У) < П) с

Moreover, if у = u — y xe H nQ — y x, then u , y xe H nQ = F ( KnQ). Thus there exist xx, v e K no such that y = F( xx), и = F(v). Since u , v e K nQ thus Q#2(x1 — v ) < l / m . We set x —v — xx, then F ( x ) = u — y x = y . For y e H no — y x there exists xe\@t2\ such that у — F(x) and Q # ( x ) < l j m .

Hence

(* * ) Л п0~ У i ^ F {xe \3S2\: Q<g2(x) < 1 / m } .

By (*) and (**),

= { У € №i \: Qax(y) < 4) c C ^ i H n Q - y i )

c C l^P |a?e \У#2\: q&2(x) < — | — Cl^ F j | .

6.4. Let 3St (i = 1, 2) be pre-(Cr)-sequences. Every complete-closed nearly-open additive mapping of |J*2| into is open.

Proof as in [⁸].

6.5. Th e o r e m. Let 3Si (i — 1 , 2 ) be prefG)-sequences and F-an ad­

ditive complete closed mapping of \3S2\ into \âSf\. Let \3S2\ be a separable metric group. I f to every p there corresponds kp such that F (\p & 2\) is of second category in [кр&г], then F is open.

Proof. The theorem follows from 6.3 and 6.4.

7. Let {pn>v) be a sequence of real numbers such as in ⁶.

00

7 . 1 . I f the metric group p вв = {C]GtTt(pn>v) , r) is complete and

71 — 1

separable in the topology r induced by the metric Qt*,p0v(°°) — lim g T*>3, (a?), then the metrics Qt*,p0v ап<% Qss are equivalent. ’ n

Proof. This follows from 5.1; Theorem 7.6 of [⁶], and from Banach Theorem on the inversion of homomorphisms (see [1]).

8. Let (p n>v) be a sequence of real numbers such that p n+lfV > p n>v for every v, ⁰ < < и < ¹ for n = l , ² , . . . and, lim p n >v < x < ¹

for every v. n

8 . 1 . I f the metric group (fi* {pn>v) ? @т*,рП1} ™ separable for

00 9

n = 1 , 2 , . . . and the metric group P У8 = ( U @*T*(Pn,v) ? T) is complete

П— 1

and separable in the topology r induced by the metric Qt*,p0 „> then G-eT.{pn,v)

= P 36 and Qt*,Pqv is equivalent to Qt*,puv f or n > N, unless GeTt(pn>v) are open-and-closed in <U ^ > 6^t* ,^p0^v) for’n = ¹ ,² , ...

Proof. This follows from 5.1, Theorem 8.2 of [⁶] and from Banach Theorem on the inversion of homomorphisms.

References

[1] S. B a n a c h , Über metrischen G ruppen, Studia Math. 3 (1931), p. 101-113.

[2] B . A. B a r n e s and A. K. B o y , Boundednees in certain topological linear spaces, ibidem 33 (1969), p. 147-156.

[3] I. K a p la n s k y , Topological methods in valuation theory, D uke Math. J. 14 (1947), p. 527-541.

[4] R. K a u f m a n , M a xim a l semicharacters, Proc. Amer. Math. Soc. 17 (1966), p. 1314-1316.

304 A. G o d z i e w s k i

[5] S. M a z u r and W . O r lic z , On some classes o f linear spaces, Studia Math. 17 (1958), p. 9 7 -1 1 9 .

[6] Z. B e m a d a n i, L im it properties o f ordered fa m ilies o f linear metric spaces, ibidem 20 (1961), p. 245-270.

[7] S. S im o n s , The sequence spaces l ( p v) a n d m ( p v), Proc. Lond. Math. Soc. 15(1965), p. 422-436.

[8] W . S lo w ik o w s k i, On the theory o f (F)-sequenees, Studia Math. 25 (1965), p. 281-296.

[9] A. O s t r o w s k i, Über einige Losungen der F unU ionalgleichung <р(х)-р(у) = <p(ssy), A cta Math. 41 (1918), p. 271-284.

[10] L. S. P o n t r i a g i n , Topological groups, W arszawa 1961 (Polish).

RO C ZN IK I PO LS K IE G O T O W A R Z YST W A M ATEMAT Y CZNE G O Séria I: P R A C E M ATEM AT Y CZNE X V I I I (1974)

A. Go d z i e w s k i (Poznan)

Errata to the paper “On the theory of sequences”

Commentationes Math. 17 (1973), p. 291-304

On page 291 in the introduction, the following sentence is not exact :

“In Sections 2 -4 we prove, using the method of the metric groups theory, that ' on the subfield Q° this function takes the form T{ • ) = \-\p, where 0 < p < oo, and it cannot appear in another form”.

This fact is true, when we have archimedean norm in the field of the rational numbers and when this norm is not non-archimedean. The function T ( ‘ ) has not such the form in the case non-archimedean norm on this field.