{X, A, p), where Ф is unbounded and satisfies the so-called condition (Z2), is not locally bounded if and only if there exists a power series

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X IX (1976)

W.

and U.

(Bonn, Federal Republic of Germany)

by power series with finite domain of convergence

In this note it is shown th a t an Orlicz space L

{X, A, p), where Ф is unbounded and satisfies the so-called condition (Z2), is not locally bounded if and only if there exists a power series

which converges only for t = 0 and t — 1.

W. Zelazko [7] shows th a t in the complex F-space $ [ 0 , 1] of Lebesgue measurable functions on the unit interval [0, 1], with identification of two functions equal almost everywhere and with topology of convergence in measure, there exists for every finite subset 1) of the complex plane C a power series

which converges in $ [0 ,1 ] exactly on Fu{0}. (For a more special result see also [1].)

Zelazko asks for suitable characterizations of F-spaces, in which there are power series with domains of convergence different from discs.

In this note we shall show th a t the non-locally bounded Orlicz. spaces [4]

L

(X, А, [л), where Ф is unbounded and satisfies condition (zl2), are characterized by this property.

Let Фр) be a continuous, non-negative, non-decreasing function, defined for t > 0, vanishing only for t = 0. We shall assume, moreover, th a t Ф satisfies condition (Z2), i.e. there is a positive constant h such th at Ф(2t) < Jc0(t).

* The authors were supported by the Sonderforschungsbereich 72 of the Deutsche F orschungsgemeinschaft.

OO

CO

] ? æ ntn, ocn€ $ [ 0 , 1], te C

Let

А,

be a totally u-finite measure space (cf. [2]). By

A,

we denote the real or complex linear space of all real resp. complex-valued, /f-measurable functions with J

{x): = f Ф(\æ\)d/л < oo. As usual we

x

identify functions which differ only on a set of measure zero.

L

{X, A, y) is called Orlicz space. I t is an F-space with F-norm

||#!|ф: = inf{e> 0: «7ф(а?/е) < г}. The sets Х ф(е): — { x e L

{X, А, y):

J

{x) < e} form a basis of neighbourhoods for the topology (cf. [3], [4]).

OO As (X , A, y) is totally c-finite, X may be expressed as X = X 0u (J a{,

i—l where X 0eA is'th e non-atomic part of X and the family of atoms (we do not consider the case when the number of atoms is finite). We

oo

set A: = U ai • i—l

Our main result is the following

Th e o r em.

A n Orlicz space L

(X, А, y), tvhere Ф is unbounded and satisfies condition (A2), is not locally bounded if and only if there exists a power series

OO

J ? x ntn, xve L

(X, A, y,), h Й resp. C, U= 1

whose domain of convergence is the set {0,1}.

P ro o f. Sufficiency of the condition follows from the existence of a p-homogeneous norm in a locally bounded F-space (cf. [1], [7]).

ISTow let

A,

be not locally bounded. Then one of the sub­

spaces

L

(A): = {xe L

(X, A , y): J <P(\x\)dy=

} or

L

(X O): — {xe L

{X, A, y): f Ф(\х\)dy = 0}

A

is not locally bounded. It suffices to construct a power series with the desired property in th a t subspace, which is not locally bounded. Therefore we may restrict ourselves without loss of generality to the cases where

A,

is either a non-atomic or a purely atomic measure space.

Because we have lim j|æj|<p — 0 if and only if lim J

(xn) = 0 [3], it

n-> oe oo

oo in

suffices to construct a power series xntn with lim / Ф ( | ^ xn\ 'jdy = 0

n —l m->oo X n = l

and

m

lim j ф{I xntn\)dy = oo for t$ {0,1}.

m - * ° ° X n =1

A. Let (А, А, y) be non-atomic. From [5] we get th a t for every ne N there is a positive real number sn with

Moreover, we can choose the sequence {sn}n£N in such a way th at

1

п—Л

1

n

{Sn) < p{X).

Therefore there exist pairwise disjoint sets FneA with y ( E n) ~~~щ—у Now we define for all ne N

У o: = 0 , Уп • ~ ®пХе п )

Я'п’ Уп У п—l )

and consider the power series

CO

a )

71 = 1

Ш

We have J ф{ xn) = J

{yn) =

/m, and so series (1) converges for t —

n= 1

For every ^ { 0 ,1 } there exists an N e N, such th a t for all n ^ N n < |(1 — t)tn\. Then we get for all m > N

- фv * -

n = N n = N n —N

7i0{S%

m—1

n = N

which implies th at series (1) diverges for t

{0,1}.

B. Now let (X, А, y) be purely atomic. We first prove the following assertion for non-locally bounded spaces Е Ф(Х, A, y):

(L) For every neighbourhood Х ф(е) there is a neighbourhood Жф(д), such that for every real A0 > 0 and any ne N there exists a function x: = (ii)uN€ Х ф(е) with

Xe S(n): = {y = (rji)UN€ L

(X, А , у): щ = 0, 1 < i < n}

and X

x

N ф{д).

P ro o f. As L

(X, A, (

) is not locally bounded, there is a <5X > 0 and for every 1 > 0 a function x e N

(s) with Ы

N ф(д1). Set <5: = and take any ne N, A0 > 0. We can choose M so large th at

(2) Ф(М)/л{а{) > e for all ie {1, n) and Xx < 20 so small th a t

(3) Ф(М'Л1) ^ --- for all ie {1, n}.

п-/л(а4)

Then for all x = (1г-)^дг€ X

(s) we have Ц{\ < M, 1 < i < n.

By assumption th a t L

( X , А, /л) is not locally bounded, there is an хеХф(е) with Xxx

X

(ô1). Prom (2) and (3) we get Xxx-% n e N

(S)

u {«d

г = 1

and therefore ЛхХ'% n

N

(ô). I t is obvious th at x% n satisfies

v\ и {«d x\\j {«d

г—\ г = 1

the assertion.

Now we construct the power series. Por every s = 1 jn, n e N , we choose ô(n) > 0 so th a t (L) holds for <S0: = 2d(n), and take t(n)e N so

m

large th a t t(n)‘ô(n) > 1. Furthermore we set r0: = 0, rm: = £ t(n ) f°r

71 = 1

all me N, and X(i): = -4— for all ie N with r rfTm

m

< i < rm.

By induction we now define for all ie N u{0} functions х{, y{e L

(X, A, /

) and natural numbers n{. Set n0: — 0, x0, y

= 0. We assume th a t xif y{, n{ have been defined already for all 0 < i < i 0. There is an me N with rm_x < i

< rm. From (L) and the definition of ô(m) we get th a t there is a function

with the property X{i

)x

N

[2ô(m)).

We choose niQ so large th a t for

(4)

0-

# -4 L 1+.W

we have X(i

)y

N

{d(m)). (4) implies th a t every two functions y{(t) and yj{t), i Фу, have disjoint support. Obviously we get limJ

{yn) = 0.

7l->oo

For every ne N and all i with rn_x - f 1 < i < rn we have 4i)Vi = -T~ Уг4 N

(ô{n)).

T n ' n

This and the construction of the numbers t{n) now implies

»* iv t \

• Ц

m A =

> * (» )•« (» )> i -

i=*rn-1+1 im,rn-1+1 n

We set z{: = yi — y

for all i e N and consider the power series

©o

I z A

i— 1 n

Clearly J ф ( 2 zг) = J<p(yn)i an(i so limJ^f y n) = 0 implies th a t the

{= 1 71— >00

power series converges for t = 1.

For every t i {0,1} there exists an Ne N such th at for all i > N we have jzi < |(1 — t)tf\. Then we get for every r»-i > N

Vn+1

rn rn

•41

2 ^, 4441

2 2

4 ^-?*)

00

which implies th a t the series diverges for t i {0,1}.

i = 1

R e m a rk . If the function Ф is bounded, the assertion of the theorem still remains true under certain conditions, e.g. when

00

(a) (X, А, y) is purely atomic (i.e. X — U af) and inf y( at) = & >

,

7=1 n e N

or

(b) (X, A, y) is not purely atomic.

In case (a) L

(X, A, y) satisfies condition (L) for sufficiently small

€ > 0, and so part В of the theorem’s proof may be used without change.

In the other case we may apply Zelazko’s result [7] to the subspace J

(Xo, A, y), where X

is the non-atomic p a rt of X.

However, for bounded Ф the theorem is not true in general. Let (X, A, y) be purely atomic with y ( X ) < oo. Then the Orlicz space L

(X, A, y) is the space s of all sequences, which is locally convex and not locally bounded.

Looking at the construction of the power series one easily sees that we may apply Lemma 1 of [7] to get the same result for non-locally bounded Orlicz spaces, which Zelazko has proved for the space S [ 0,1].

Corollary 1.

A complex Orlicz space L

(X, A , y), where Ф is un­

bounded and satisfies condition (zl2), is not locally bounded if and only if

OO

for every finite subset F a C there exists a power series 2 ænfni which con-

71 = 1

verges if te D u{0} and diverges i f ^ R u { 0 } .

As in a locally pseudoconvex space convergence of a power series

oo

£contn for tQ> 0 implies convergence for \t \ < t0, we also get for an un- ns» 1

bounded function Ф a result analogous to Theorems III.3.6 and II I .3.7 in [6].

Corollary 2.

A n Orlicz space L

(X, A, p), where Ф is unbounded and satisfies condition (A2), is locally bounded provided it is locally pseudo­

convex.

References

[1] L. A rnold, fiber die Konvergenz einer zufalligen Potenzreihe, J. Reine Angew.

Math. 222 (1966), p. 79-112.

[2] P. R. H alm os, Measure theory, Van Nostrand, London-New York 1950.

[3] J. M u sielak and W. O rlicz, On modular spaces, Studia Math. 18 (1959), p.

49-65.

[4] W. O rlicz, On spaces o f Ф-integrable functions, Proc. Intern. Symp. on Linear Spaces, Hebrew Univ. of Jerusalem, 1960, p. 357-365.

[5] S. R olew icz, Some remaries on the spaces N (L) and N (l), Studia Math. 18 (1959), p. 1-9.

[6] — Metric linear spaces, Monografie Matematyczne, Tom 56, Warszawa 1972.

[7] W. Z elazko, A power series with a finite domain of convergence, Comm. Math.

15 (1971), p. 115-117.