71
Internat. J. Math. & Math. Sci.
(1986))1-79
Vol. 9 No.
ON SOME SPACES OF SUMMABLE SEQUENCES
AND THEIR DUALS
GERALDO SOARES DE SOUZA and G. O. GOLIGHTLY
Department of Mathematics
Auburn Unlversity
Alabama 36849
(Received May 18, 1984 and Revised March I], 1985)
Suppose that
ABSTRACT.
n>oSUp aj Jl
lla’S
and
aJBl S "a"SllSl’J
S
is the space of all summable sequences
J
the space of all sequences
this inequality leads to
e
a
wlth
of bounded variation
6
e
description of
dual space
It, related Inequalities, and their consequences are the content of
paper, In parttcular the Inequality cited above leads directly to the Stolz
of S as J.
this
form of Abel’s theorem and provides a very simple argument.
Also, some other
sequence spaces are discussed.
KEY WORDS AD PHRASES.
Dual spaces, sequence spaces, bounded operators.
1980 Mathematics Subject Classification:
I.
INTRODUCTION.
S
Suppose
is the space of all summable sequences
Ls
In>0
J--
"s"s’Sup
norm glven by
An example of such an
s
(-1)n
n
46A45, 46A20
n
O, I, 2, 3,
so that
Here, as is readily verified,
2.
with
after
s
n-terms.
s, summable but not absolutely summable, Is given by
X --’----’--(-1)n
=0 n+l
the natural log of
(s)(R)
n nmO
the remalnder In the sum of
n+l
series.
s
,s,
S
sup
n>O
Is an alternatlng convergent
<->
l-
in 2, the
-n
’J
This paper is based on the observation that
S, with this
norm, is naturally isomorphic to co, the space of all sequences having limit zero
endowed with the supremum norm. Moreover, if we consider S on Its own, we get
several Interestlng results.
the dual of
S Is isomorphic to co and
41, the space of all absolutely summable
S*, the dual of S, is isomorphic to 41
For example, since
co, is Isomorphic to
sequences, then It follows that
co*
72
G.S. DE SOUZA and G. O. GOLIGHTLY
However, if we compute the dual of S without using the Isomorphism with c0, we
find that S* is isomorphic to J, the space of all sequences 8 of bounded
variation with
11311j---11301
8
by
n
Bn- 8n_ll
n.l=
+
=
an example of such a
O, I, 2, 3,
n
so that
n+l
J
Of course,
t
is isomorphic to
8
being given
i!n 1...]
n+l
+
II J
n=l
Another example is that we get the following
I.
I
s 8
< ,’s,, S
j=o
used to glve a proof of Abel’s Continuity Theorem.
8 e J
If
new inequality:
and
S
s
2.
H
Similarly, we consider the space
which can be
"8"j,
(Xn):_0
X
of all sequences
such that
n
J=0 j[
"H n>0Sup
{l[l
that
I.
H
%n (-l)n’
is finite; for example
and only if
XOp +
(8)
n
n=l
(0
8 n-1
+
Consequently we have that
for
lim 8
0
e J*
If
J
8
Also, we glve characterizations of the bounded linear operators on
obtained from a description of operators on
so
A e H, 8 e J, and
We get the following inequality: If
1400 +n--1 n(O- n_l)[ _< ,,"H.,Sllj.
then
O, I, 2, 3,
n
c
S
easily
These results, however, may be
obtained using the uniform boundedness principle.
To make the presentation reasonably self-contalned, we shall include a resume
of pertinent results and definitions.
PRELIMINARIES
2.
DEFINITION 2.1.
Let
(X,
fix)
.
and
(Y,II
y:X
R
by
y(X)
n=O
b two sequence
for those x
XnYn
spaces, which are
(yn)
Y define the
(xn) in X
for which the
For a fixed
Banach spaces with the respective norms.
mapping
y)
y
series converges.
..lY(X) <Mllxll x
Assume that
y
M
with
is a bounded linear functional on
X
space of all bounded linear functlonals
IIx.
y.
b(y)
sup
x
X
If
the dual of
[(x)]
and
b
Is onto and
X, in this sense
DEFINITION 2.2.
some absolute constant, that is,
for any fixed
we rlte
llyllx,
Y
Y
to
X*
S, I,
,
Y
s
(s)
n n=O
summable sequences such that
H
Co, and
CO
{
(n)n=o’
(Yn)=O
lls
p>_.O J
n n=O
{
=0
I1(R)
lim
n+
sup
< (R)},
J>...o
Yn
0,
,,Y,c0
sup
n>O
be the
is
by
S
X*
given by
then we say that
X*.
We define the sequence spaces
Let
X, endowed with norm
on
the mapping from
llylly
Y.
y
p
respectively
SPACES OF SUMMABLE SEQUENCES AND THEIR DUALS
n=O’
n
H
n>O
J=O
We believe that this is the first time that
TqF)REM
2.3.
T:S
The iometry
-]
S
The space
c
2.4.
:),I’IN[TION
S
is taken as a space with this
is a Banach space isometrically isomorphic to c 0.
T(s)
(r (s))
and T-I :CO
S by
being given by
(n- n+1)n=O
()
73
n=O
n
r (s)
where
(X,I
Two Banach spaces
fIX)
(Y,U
and
fly)
are isomorphic
eq,Ivalents if there exist a one-to-one and onto lnear mapping
Nil
,IX
--< ITXlly _<
such a
T
Mllx;l
with
X,
I;Xllx,
llTxlly
with
N
M
and
absolute constants.
then we say that
X
Y
and
T:X
Y
such that
If there exists such a
are isometrically
ono rphlc.
Notice in Theorem 2.] T and T -I can be represented as infinite matrices,
0
0
0
0
0
0
-I
0
0
0
TIEOREM 2.5.
M
0
0
0
0
0
0
-I
-I
0
0
-I
0
0...
is a bounded linear operator from
c
is represented by an infinite matrix with columns in
uniformly bounded in
i)
llm
n/
rank
0
i l-
That is if
for any flxed
M
(rank)
into
c
c
if and only if
and rows in
il
then
k,
n>O k=O
THEOREM 2.6.
$
that for any
such
(n)
() e
n
co*
is in
I
()
e c
and
if and only if there is a
JltJ
co*
n=O nYn’
IIJJ
I
6
(6)
e
n
I
such
in which case there is only one
M
G.S. DE SOUZA and G. O. GOLIGHTLY
74
I
L-I:I
O, and
8-I
(Bn)
I
L:J
The isometry
space, isometrically isomorphic to
)
(B
is given by L(B)
n-I n=O
n
is a Banach
J
The space
THEOREM 2.7.
J
I
t-l()
by
n=O
j=O
with
e
(5)
n
I
and
J.
E
The space
THEOREM 2.8.
H
space isometrically isomorphic to
is a Banach
E(R).
n
:H
The Isometry
by
-l(U)
3.
THE DUAL OF
(n- n-1)n=O
S
For
N
>
s
Bn n
N-I
nO=
r (s)
where
with
(Holder’s type inequality)
THEOREM 3
Proof:
()
is given by
follows that
.
.
s
(s) e S
n
and
B
()
n
J
I, we notice that
N
rn(S)[Bn__l Bn-l]l IrN(S)BNI IrN(s)I’IBNI
-r0(s)B
n
Now since
s
rN(s)
n=O
If
0
BN
.
.
N
as
r0(s)B0 +
Sn n
B
is summable and
s
Therefore we have proved that
rn(S)[B n Bn_l]
n=1
Is bounded, It
consequently
so that the theorem is proved.
Consider the mapping
(8 n)
8:S
R
In
J.
is a fixed element
s
where
Is
At this point, a natural question is:
S.
a bounded linear functional on
(s)
B
n=.O n n
In view of Theorem 3.1, we have
defined by
are these all the bounded linear functlonals on
S?
The next result tells us
that the answer is yes.
,
(Duality Theorem).
THEOREM 3.2.
such that
IIIS,
,J
Prof
lllj.
Se
(s)
(s)
Sngn,
e
Bn
for any
s
e S
(s)
n
()
Moreover
s 8 then 0 e S* so that the mapping
n n
n= 0
is an Isometric tsorpht fro J t S.
then e already hae seen by Theor 3.1 that
e Se. So it reatns t pre the
S, that ts
Se; then using Theore 2.3 e hae
a bounded linear guncttonal on
Let
rl
then there is a unique
(s)
(a)
by
n0
first part.
s
n--0
Conversely If
degtned
If
that is,
S
If
SPACES OF SUABLE SEQUENCES AND THEIR DUALS
o T
Now notice that
-I E
75
(Yn)
co*, so that there is a Y
l (see
Theorem 2.6)
such that
(s)
I Yn
T-l)[(nSj)
J=
( o
[_ n s
n=0
n=0
and
A
o
T-II c0*
,yU
(3.3)
n
Observe that (3.3) can be written
8
where
j0Y_L-I(y),
n
8
Since
8
J
g
such that
inequalities we
(8)
by
SB
ve
We
8"
lBtIj
ve
(B n)
therefore we may let B
Consequently given
E
there Is a
S*
c 0*
n=0
I, It follows
tt*tIS*
implies
J.
In
is
y
On the other hand Theorem 2.6 tells us that
=0
liT-lit
Now as
B
n=0 Snj =0 j;
(s)
(since
ttOS*
that
Also notice that Theorem 3.1
liJ iOS*"
CB). Therefore putting together
.I; consequently the mpplng $:J
these last
S*
o
defined
Is an Isometry, so that the theorem Is proved.
used Theorem 3.1 to characterize all bounded linar functlonals on
Now we are going to use this theorem again to give a proof of Abel’s continuity
S.
theorem which Is as follows.
THEOREM 3.4 (Abel’s continuity theorem).
f(z)
n=0
then
>
N
.
C
p=NapzP _< (I+C)(ap)NS
0
N-I
A
<
N
/2.
is summable and
is restricted to approach the
a
zPl
<
,(z p(R)
)p.Nllj-,(ap)N,S
by the above inequality applied to the sequences
p=O P
so that
Hence
+
N
as
(a) and
p
(zP),
+
+ C.
since
(an)
Consequently,
e S.
Then
N-I
P
B
#z[
(zP)N J
N using the hypothesis we have
fix
z
(an)
be a positive absolute constant such that
+
P
a--
llm f(z)= f(1), where
z+l
’irst of all, let
Proof:
for
anZ
If
p=0 P
(2+C)](ap)N[ S
a z p=
a
lira
p
z+l p=O
p=O p
p=N P
/2.
For
z-1
sufficiently sll,
Therefore the theorem Is proved.
G.S. DE SOUZA and G. O. GOLIGHTLY
76
J
THE DUAL OF
4.
(Sn)
s
Of course, for fiel
S
in
(8n)
8
!fete
lira 8
I
.
nZl
+
B_1)
l: s
0
where
jfoSj)p + .jfoSj(Bj-o)
for
Z j --s_ I.
S.
.
e J, and
(B)
n
(L) e H, B
n
L
If
[LOP + n:lT L(pn -8n-l)[
then
and
0
is in
so that
,s(B)
then
0
< [LH.,B,, "J
Ths inequality follows easily from the observation that
Proof:
10P +
with limit
(Holder’s type inequality)
TIiEOREM 4
}LOp
J
is in
l: L(p
Z=0 0;
0O +
p
As a key to
I*.
members of
j.OSjBj
Os(g)
the formula
However, these are not all the
our description of J*, we note that if
J.
defines a bounded linear functional on
oB0 +
L (p- 8
n -I
n
Lj)[8n
n=l
8n
n
ZOLj[(IB01 +
sup
Xn(p- Bn_l) _< n>0
n=l
J=
n
so that
!llBn- Bn_l}).
S, we consider the mapping
As we did for
I];
CL:J
R
defined
Z
(L)
is a fixed element in
L (p
where L
B n-1
n
n
nil
is a bounded linear functional on J.
Theorem 4.1 assures us that
by
Lop +
eL (8)
H
L
The claim of the next theorem is that these are all the bounded linear
J.
functionals on
Here, our representation is in a slightly different sense
than that of the Definition 2.1, the latter appearing more natural to us.
(Duality Theorem).
THEOREM (4.2)
L
8
(Ln)
(8 n)
[{
e J.
Moreover
Proof:
Here,
J*
then there is a unique
that is, (8)
L
ll@llj.
:H
So that the mapping
morphism.
eL’
such that
e J*
If
X00
Conversely if
H.
(L)
defined by
+n=l@[" Xn (O-Bn-l)
llm 8.
0
By the observation preceding the theorem we see that it remains to
e J*; then in fact using Theorem 2.7
Let
-!
L-I)(L(8))
(L o L(8))
(@ o
(@ o L-I)((8
8n_l)nffiO) 8-1 O.
o L
Now notice that
(4.3)
e: J*.
is an isometric iso-
prove the first part.
@(8)
then
@X
for any
(
(8)
o
-I e
n
I’
so that there is a
L-1)((Bn Sn-1 )-n=O)
n
By Theorem 2.8
consequently
U
n
(4.3)
OXj
for
n
(8)
becomes
>
0
.
n=O
[ Un(Sn
L )(8
J=0
N
8
n-I
L
and some
LOB0 +
n
(u n) e
.
8n-1 )"
n.O
Observe now that
N
n
I I Lj)( n
n=0 j=O
p
(Xj)
e H;
-B n-I
n
I I Lj)(8n
n:l =0
8 n-1
such that
.
SPACES OF SUMMABLE SEQUENCES AND THEIR DUALS
N
X0 N +
n--I
Xn[N
X0B N +
8n_l
N
[B N
100
n--I
Xn[N
0
0]
k
n= 0
n
+
n--1
kn[0-Bn
]"
.
[B N
since
p]
lOp +
I
0
II
H
ln[P
8
o L
IIIH.
llm 8)
n
[-
,,
N
-I
Therefore we have
n-I
-IIII*
<
hand Theorem 4.1 gives us
llIIj, _<
n=l
(p
N
as
II
Again by Theorem 2.8
lllll
n-I
n--O
n=l
or
0
Taking the limit as
,op +
_)(,,- ,_)
J =0
+
N
we get
n=O
N
77
N
L-III<
o
ll’tIl3**llL-111
that
(B)
n
llull.
n
sup
n>O
sup
llllj, sinc IlL-Ill _< I.
S llX"H""j(since
X
),
Putting together these two inequallttes we have
J=O
On the other
which implies
llj,
so that the theorem is proved.
S
CHARACTERIZATION OF OPERATORS ON
5.
In this section we characterize all bounded linear operators on
S, even
though an alternate proof can be given using the uniform boundedness
We rather use the characterization of operators on
principle.
cO
In fact
we have the following result.
THEOREM 5.1.
A:S
The linear mapping
is an infinite matrix
(ank)
cO
is bounded if and only if there
such that for any
ao0
alO
aOl
all
a02
a12
a0n
aln
ano
ant
an2
ann
s
(Sn)
S
SO
AS
satisfying
0, for any fixed k
i) lira a
il)
sup
n>O k=O
Here
0
for
n
Note that the operator
Proof:
maps
an(_l
lan(k_l)- ank
co
into
co, that
is
>
0 i.e. the columns of the matrix are in c 0,
<
i.e. the rows of the matrix are
uniformly bounded in J.
0, I, 2, 3,
where
ATT-I A
P
P:c0
S
T-
Is as in Theorem 2.3
c0, so that applying Theorem
2.5 we
get the desired result.
THEOREM 5.2
The linear mapping
B:S
S
is bounded if and only if
represented by an infinite matrix such that for some
A:S
c
B
we have
is
,
78
G.S. DE SOUZA and G. O. GOLIGHTLY
X
Is defined by
B
A-
(
is a shifting of
TB
S
co, where
wlth
Is as in Theorem 2.3.
T
Now using the matrI representation for
-1
I
0
0
0
0
0
-i
0
-I
0
B
ank a(n+l)k.
ank .;=nbjk
Note that
T-I
we get
i]A
[I
A
IA-
B:S
Notice Theore 5.2 is equivalent to saying that
B
r(B)
then the rows of
,
r(B)
r(B)
with
(Ynk),
are uniformly bounded In
The linear mapping
C:e
S
S
Notice
c0
C
T-1M
therefore
T
c0,
where
I
up by one row.
Is a bounded
B is summable,
S
where
b
Ynk
Jfn
k’ then
J.
is bounded if and only tf
M:c
represented by an Infinite matrix such that for some
C
MM is a shifting of M up by one row.
Proof:
A-
is an infinite matrix such that each column of
and if we consider the matrix
THEOREM 5.3.
A-
Is formed by the shlftlng of
is the Identlty matrix and
operator If
That is,
T-I.
B
so that
X (nk)
up by one row)
A
A:S
We define
Proof:
A
where
as in Theorem 2.3.
C
is
we have
e
So define M
TC;
M.
rl
are special A’s as are the B’s. The C’s are
B’s. The reader might wish to compare these last three theorems with
Exercises 45 and 46, pg. 77, of [1].
We would like to point out that the spaces tt and J may also be found in [1]
(pg. 240). Another space there, denoted by cs, is the space of all
M’s
Note here that the
special
n
summable sequences
(sj)
s
.
normed wlth norm
on
shown, as pointed out there, that es* looks llke J.
,I
that
is not the dual of
cs
<
Ilsn
the inequality
Let
s
so
I,
(s)
BO= O, BI
s3
s4
-I, B2
<
In the sense of Definition 2.1.
does not hold
.ll,l
[max
n>0
-2, 83
ZO
J--
s
-2, s6
0, s5
S
and
llsll
For
>
I]-,,BII J
-4, B5
-3, B4
_< 211SNcs.
0
let
ej
So that
(B):
n
O; also, B
-4.
B6
does not hold.
are equivalent Banach spaces:
S
Consider the example:
s7
To Illustrate further the difference between
and
In particular,
be given by
s2
sl
s.8
ffiOS [.
It may be
n>0
We hasten here to note
H, max
if
Is In
be the sequence
S
T
.
s
and
cs
Is In
S
If and only if
such that
T
S, we note that cs
ns!
then
@
<
cs--
2|s!
S
Is In cs*.
and If
k
Xk
O.
79
SPACES OF SUMMABLE SEQUENCES AND THEIR DUALS
Now for
S
[__08js]
(s)
by
(ej))
8j
cs* with the sequence
is in
then
i,j
{(e)}
In
4.
J
J
and
is defined on
So the tural
Is not an Isometry.
J, obtained by noting that cs
sequences normed with
isomorphic to the space c of all convergent
represenbeing isometrically isomorphic to I. This
Isomet from cs*
isometrically
;
the
tation of c*
as
no In
(so that
In
assoclatlo of
There ts an
{(ej)}j=0
the sequence
But if 8 is the sequence of the example and
IS*.
I(%(ej)}nj
’)i=0sj(ej);
S*,(s)
in
onto
c*
dual
Is complicated by
and also the representation of cs* as J,
floweret,
c.
In
span
does not ve dense
the fact that the sequence
as J) Is de somewhat simpler by applying
cs*
(of
the latter representation
N
the Nth partial
Theorems 2.7 and 3.5. For s in S let
I,
{e]}
sum
with
o
o(s)
s, and
L
j(s)u0 +
lm
0
uses (as
-
(ej) 8j. Now,
we ve (s)
=oS(e) s]
0(> + l=[() s_(>){ n Bn-1
B n-1
lm B, u n
Bn); so v L(U), wth
=OSj.
s In Theorem 2.7,
l=
00(> +
Z=I
B
(){
Sn_l(S)Vn
n
n
n [I))
should be
gven
used there
s
lm B +
(v
B0
the space
by
mlB,l
SN(S)
cs
Let
Bn
+
be In
cs*
wlth
These computations suggest that f one
used n J
rather than S the norm to
(lira B( + Zn=l(Bn- Bn_l(-
the same as ours (see
I
the Holder’s pe Inequality we get
s
pg. 239).
(Z=OBj(
However, the norm
With ths norm,
<
iI’
,B,
NBN, on J
n s
and
of ths last paragraph wth some
The reader might wish to compare the remarks
and page 964 of [3].
of D. J. H. GarlIng In pages 999-1000 of [2]
certain power seres spaces ntroduced
on
norm
These ideas were suggested by a
[4 of Fourier sne seres and Ldstone
by the latter uthor In an nvestgaton
series.
REFERENCES
[I]
DUNFORD, N. and SCHWARTZ, J. T., Linear Operators, Part I, Interscience,
New York, 1958.
[2]
GARLING, D. J. H., On Topological Sequences Spaces, Proc. Camb. Phil. Soc.
63 (1967), 997-1019.
[3]
GARLING, D. J. H., The B and y-duality of Sequences Spaces, Proc. Camb.
Phil. Soc. 63 (1967), 963-981.
[4]
GOLIGHTLY, George O., Sine Series on [O,Z] for Certain Entire Functions
and Lidstone Series
To appear in J. Math. Anal and Applica.
[5]
ZYGMUND, A., Trigonometric Series, Cambridge University Press, London, 1959.