Packing constant of a type of sequence spaces*Abstract. In this paper, the packing constant for a general type of sequence spaces is

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

Tingfu Wang (Harbin) and Youming Liu (Linfen)

Packing constant of a type of sequence spaces*

Abstract. In this paper, the packing constant for a general type of sequence spaces is discussed, and a uniform and simple formula is obtained. A number of known results, e.g. the packing constant of l2, lp (p ^ 1) and lM (equipped with the Orlicz or Luxemburg norm), are corollaries of this formula.

The packing constant A(X) of a Banach space X is defined as follows: if r ^ A(X), then an infinite number of open balls with radius r can be packed in the unit ball B(X), and if r > A(X), only a finite number of open balls with radius r can be packed in B(X). To be more precise,

A{X) = sup{r > 0: 3{xf} œ B(X),

llxj ^ 1- r , t|xf —x^-ll ^ 2r (i,j = 1, 2, i ф])}.

The knowledge of A(X) is of great benefit for studying the structure, isometric embeddings, noncompactness and reflexivity of spaces.

For the sequence spaces l2 and lp (p ^ 1), A(l2) and A(lp) have been found by Rankin, Burlack and Robertson [1], [7]. In 1970, Kottman [4] showed that

3 ^ A{X) ^ \ for any Banach space X. Cleaver [2], in 1976, found upper and lower bounds of A(lM) for the Orlicz sequence space lM equipped with the Orlicz norm, under some condition. Afterwards Zaanen showed that under Cleaver’s condition, lM degenerates to lp for some p ^ 1. In 1983, Yining Ye [9] obtained an exact expression for A{lM) for lM equipped with the Luxemburg norm. Ting­

fu Wang [8] got rid of the restriction given by Cleaver and obtained an exact expression for A(lM) for lM equipped with the Orlicz norm, but it is very com­

plicated. In this paper, we are going to discuss a type of more general sequences containing all above special cases, and to obtain a uniform and simple formula for the packing constant. All the above results are corollaries of our Theorem 2.

By [3],

A(X) = ^ D^X] 4, where D{X) = sup inf ||x(n)- x (m)||.

2 + D(x) {*<">} <=S(X) пФт Obviously, we only need to calculate D(X).

This project is supported by NSFC.

Denote the, sequence (x1?x2,...) by x = ( x j f . Let e(l) = (0, 0, 1 ,0 ,...) with 1 in the ith place (i = 1, 2 ,...). If x = { x jf , y = {>>;}*, we set xoy = (xl5 yl5 x2, y2, ...)•

Th e o r e m 1. Let [X , ||* ||] be a Banach sequence space with the following

properties:

(1) e(l)e X (1, 2, ...) and {e(l)} f is a Schauder basis in X.

(2) ||*|| is a montonous and symmetric norm [8], i.e., x = Z ^ ! х,-е(1), У = t r = i y i e(i) and |xf| ^ \yt\ (i = 1 ,2 ,...) imply ||x|| ^ ||y||; x = Z«" i X ^ e X implies (*) and ||Zi*i xfe7t(,)|[ = ||x||.

(3) For any £ > 0 there exists a positive integer N £ such that the number of integers к satisfying ||Zf=Qk + i xie(,)|| ^ 8 ls 1ess than N E for any xeB(X) and any increasing sequence {<2fc}î° of natural numbers.

Then

D(X)= sup inf ||x(n)ox(m)||.

{ j C l J c S f i ) пФт

P ro o f. For any {x(n)}.f c= S{X) and s > 0, we shall prove the following assertions: there are a subsequence |у (и)} of {x(n)} and an increasing sequence {Qk}f °f natural numbers such that

(i) Il Z (у1т)- # ) е(,)| < h 0 ' = 1 , 2 , .

i = 1 ..; n,m ^ j ) ,

(ü)

0 0

II £ j4n)e(i)|| ^ e ( 7 = 1 , 2 , . i = Qj + ¹ + 1

n ^ j ) ,

(iii) ‘ Il Z y \ n)e (i)\ \ < e O'= 1 , 2 , . . i = Qj + 1

•; n >j).

By the boundedness of x f } ( n , j — 1,2,...)., we can use a diagonal procedure to select a subsequence of {x(n)}f (for simplicity we do not change notation) such that lim,,-** xÿ0 = x;- (j = 1, 2, ...).

For some fixed Qx ^ 1, take N t > 1 such that n , m ^ N t implies

|E P ii(x |n)- x i m))e(l)||< i £ . Let y(1) = x(Nl).

Take Q2 > Qx satisfying ||Zi“ q2 + i yi1)e(,)|| < e. Take N 2 > N 1 such that n , m ^ N 2 implies ||Z fi! (x|n) — x|m))e(l)|| < js. Let y(2) = xiN2).

Take Q3 > Q2 satisfying g3 + i у|2)е(1)|| < £• Take N 3 > N 2 such that n , m ^ N 3 implies ||Z?=i (*«") —*!m))e(,)|| < i £- Let y(3) = x(Ns), etc.

In this way, we obtain a subsequence {у(и)}® of (x(n)}“ and a sequence Qi < Ô2 < • •• of positive integers satisfying (i) and (ii).

I1) {я(0} “ is any arrangement of the natural numbers.

We prove (iii). We consider the set

J = { j : l n > j , || £ yfe^W ^ e}.

i = Q j + i

It is enough to prove J = 0 . If J ф 0 and supjeJj = oo, then we choose

Л < h < ••• a n d n k >Л (/c = 1, 2, ...) such that Il i j r ’* !

i = Q J k + !

For an arbitrary positive integer p, by (i)

\ yjMe«> II^H £ M”‘^ ,i,N I I

f - Q j k + 1 i ~ Q j k + t i - Q j k + 1

Q j k +1

> e - \ \ E (y|"p) — Уг"к))с(г)||

i= 1

^ £ - i e = ie (fc = 1, 2, p).

As y(nj,)e£(X), the arbitrariness of p contradicts the assumption (3) of the theorem.

If J ф 0 and supjgj j = j 0, by substituting for {y 0)} f and { Q j} * , we consider the new subsequence {y,0)}î° and new increasing sequence {Q'j}? of natural numbers, where y'(j) = yUo+j), Q) = Qj0+j {j = 1, 2,...). Then {у,(Л}“

and {Q'j}f obviously satisfy (i)—(iii).

Hence for any n, m, n < m,

lly,”)-y(m,||

i = l i = Qn + 1 i = Qn + 1

00 00

+ E y!n)e(i)- X y (im)e(i)||

i — Qn + l + 1 i = Gn + l + 1

Qn+ i oo

^ || E У£П)^(,)— E у!т)е(,)|| + 3e

i = Qn + 1 i = Q n + i + l

+1» . Oil+ 3e

So

< ||y(n)oy(m)||+ 3e.

inf ||x(n)- x (m)|| < inf ||y(n)- y (m)|| < inf ||y(n)oy(m)|| + 3e

пФт пФт пФт

^ sup inf ||y(n)oy(m)||+3в.

{ j i ( n ) } c S ( X ) пФт

By the arbitrariness of e,

inf ||x(n) —x(m)|| ^ sup inf ||Уп)оУт)||,

п ф т 0 » ( " ) } c = S W п Ф т

therefore

D(x) ^ sup inf ||y(")oy(m)||.

{,,(« )} < = S (X ) п Ф т

Conversely, for any {у(и)}® c S(X), let

X(1) = ( У 11, 0 , уЧ \ 0 , J 0 , У,1», 0 , У?», 0 , У ?1, 0 , У7‘», 0 , . . . ) ,

У2’ = (0, У,2», 0, 0, 0, ^{Z i \} 0, 0, 0, Уз2», 0, 0, 0, yf>,...), x<3' = (0, 0, 0, УА 0, 0, 0, 0, 0, 0, 0, y23), 0,0,...), etc.

Then ||x|n» -x (”>|| = ||У">оУ">||. So

sup inf ||yn)oy(m)|| ^ sup inf ||x(n) —x(m)|| = D(X).

( у ( " ) ) | : Э Д п Ф т {*<")}<= S(X) п Ф т

The proof of Theorem 1 is complete.

Th e o r e m 2. Let [X, ^||-||] be a Banach space which, besides conditions (1)—(3) of Theorem 1, satisfies the following condition:

(4) for any {yn)}f c= 5(20, inf ||У")оУт)|| ^ sup ||У”)оУ")||.

п Ф т n

Then D(X) = supxeSm ||xox||.

P ro o f. By (4) and Theorem 1, D(X) ^ sup^.6SW ||xox||. Conversely, for any x eS (2f), let

x(1) = (xl5 0, x2, 0, x3, 0, ...), x(2) = (0, x 1? 0, 0, 0, x2, ...), etc.

Then

||xox|| = inf ||х(и)ох(т)|| ^ sup inf ||x(n)ox(m)|| = D(X).

п Ф т { y n ) ) c S ( ï ) п Ф т

So supxes^) ||xox|| ^ D(X). Theorem 2 is proved.

Now, we explain why all conditions in Theorem 2 are satisfied for separable Orlicz sequence spaces lM both with the Luxemburg norm || • ||(Af) and the Orlicz norm ||-||M.

Since M(u) satisfies the A2 condition, (1) holds, immediately. (2) is trivial.

Also, since M(u) satisfies the A2 condition, for any e > 0 there exists b > 0 such that ||у ||M ^ e implies I M{y)> b.

Hence |E?=ei + 1 *,е(0||м ^ e implies £?=& + 1 M(xf) ^ b (certainly, E f t e i + i ^ l U ^ £ imPlies + i M (x«) ^ Ü t00)- But М м < 1 implies IM(x) < 1. So the number of positive integers к satisfying Xf=e£+i M(xi) ^ b is

less than [1/5]+ 1. Consequently, the number of positive integers к satisfying IIESoUiV I Im ^ £ (or IE?=&Uixi*(i)IU) ^ e) is less than [1/5]+ 1 for any x e B ( X ) and any increasing sequence of natural numbers.

Next, let us check (4) for the space [ZM, ||-||(M)]. Assume that x, yeS{lM), l|xox||(M) ^ ||yoy||(M). Then

xox

So, 21_M

xox

l|xox||(M) A 5

(M) || XО X || (jvf) (M)

\ / ( xox ^

^ 1, i.e. IM

J

4

^ 1.

xox ( M ) ,

<

Similarly /мСу/||х°х||(м)) ^ 2- Hence

M xoy

xox (M) ⁼ 1_M

xox

^ )+4

This means ||xoy||(Af) < ||xox||(M), which yields (4), obviously.

The case of the space [ZM, ||-||M] is more complicated. Let {x(n)} c S(lM), d = sup„ ||х{п)ох(п)||м . If d = 2, then (4) is trivial. So we may assume d <2. Take k„> 0 such that

1 ^ x(n)ox(n) _ 1

m k n 1 + 1м[ Кx(n)ox(n)\ \ 1

И = 1 ( 1 + 2/ Ml K ~ J {n = 1 ,2 ,...). Hence I M(knx(n)/d) ^ (fc„-l)/2 (и = 1 ,2 ,...).

If l i m „ = oo, we may assume l i m „ ^ = oo. By f c .- l > I_M k„x(n)

> /с„х(п)

- l = “f - l .

m «

taking n-> oo, we obtain d ^ 2. This is a contradiction. If lim ,,-^ ,, < oo, we may assume lim„_o0/cn = k0 < oo. So

x(n)ox(m) 1 Д T / , x(n)ox(m)

^ Г 1 *o

M \

- к 11 + /м

^ ! И

- / M

0

k„x(n)

d k„x(n)

1

[M

- / M d kmx(m)

+ 1_M kmx(m) + 1

I/ v*(*0\ / Is y(«)

M к Y(n)

IV rj •A'

+ / ^“0 •

d

1 / fc- 1 /cm — 1

! 1+ - ^ - + - ^ - + / M' d

k0x(n)

k„x(n)\

d )■ + J M h уШ

IVл Л

A f

k„x{m)

2 ' ~M\ d

►1 (и, m-+ oo).

(Here, we have used Lemma 5 in [3]: if M(u) satisfies the A2 condition, then for every e > 0 there is a Ô > 0 such that I M(x — y) < Ô implies \IM(x) — IM(y)\ < г for any x e l M and yeB{lM)). Therefore, inf„*m ||x(,,)ox(m)||M ^ d.

Now, applying Theorem 2 to the space [ZM, ||-||M] we get D{l M) = sup ||xox||(M) = sup infjc > 0: / J — J ^ 1 >

Il JC 11 (A4-) — 1 | |x |i ( M ) = l L \ C J ^J

= sup inf{c > 0: I M{x/c) ^ %}.

||*II(M) = 1

By the A2 condition, in f{ c> 0 : IM(x/c) ^ = cx, where cx satisfies IM(x/cx) = j. Hence we obtain

Corollary 1. D ( lM) = sup {cx ^> 0: I M{x/cx) = { } .

11*11 (JVf) = 1

This is the main theorem in [9].

For the space [ZM, ||*||M], D{lM) = sup ||xox||M =

II*IIm = i

sup inf | ( l + I M(k(xox)))

| | * | |m = i k > o k

= sup inf 1(1 + 2I M{kx)).

11*11 m ~ 1 k > 0 К

By the A2 condition, for fixed x gS(Im) and к > 1, there exists a unique dx,k ^> 0 such that IM{kx/dXjk) = (k —1)/2. As к increases from 1 to oo con­

tinuously, к/dXtk must increase from 0 to oo continuously. So D(lM) = sup in f i( l + 2 I M(hx))

II*IIm = i h > o n

sup inf

*11 м = i fc>i d-x,k

к ^{1+ 2 /}Af kx

*-x,к

= sup inf + 2 ^ — - ^ 1 = sup infd**.

l l * l |M = i k > i I к \ 2 ) ) n * | | M = i k > l

Hence we obtain

Corollary 2. D ( l M ) - sup inf

* | | м = 1 k > 1

d x , k > 0 - / M

kx dx,k, This is the main theorem in [8].

D(lp) (p > 1) may be obtained from Corollary 1 or 2. For example, using Corollary 1, we obtain

D{lp) = sup

ll*lli =i I i= 1

= U = 21/p (p > 1).

References

[1] J. A. C. Bur la ck , R. A. R a n k in and A. P. R o b e r ts o n , Packing spheres in spaces lp, Proc.

Math. Assoc. Glasgow 4 (1958), 145-146.

[2] С. E. C le a v e r , Packing spheres in Orlicz spaces, Pacific J. Math. 65 (1976), 325-335.

[3] A. К am in ska, On uniform convexity o f Orlicz spaces, Indag. Math. 44 (1982), 27-36.

[4] C. A. K o ttm a n , Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150 (1970), 565-576.

[5] M. A. K r a s n o s e l ’s k ii and Ya. B. R u tic k ii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen 1961.

[6] J. L in d e n s t r a u s s and L. T z a fr ir i, Classical Banach Spaces I, Springer, 1977.

[7] R. A. R a n k in , Packing spheres in Hilbert spaces, Proc. Math. Assoc. Glasgow 2 (1955), 139-144.

[8] T. W an g, Packing constant o f Orlicz sequence spaces, Chinese Ann. Math. 8A (1987), 508-513.

[9] Y. Ye, Packing problem in Orlicz sequence spaces, ibid. 4A (1983), 487-493.

[10] A. C. Z a a n e n , in: Lecture Notes in Math. 945, Springer, 1982, 263-268.

DEPARTMENT OF BASIC SCIENCE, HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY HARBIN, HEILONGJIANG, CHINA

DEPARTMENT O F MATHEMATICS, SHANXI NORMAL UNIVERSITY LINFEN, SHANXI, CHINA