Publicacions
Ma emá iques,
Vol
36
(1992),
167-173
.
A
bs ac
A
p oblem
which
is
equen ly
encoun e ed
in
analysis
is
he
in e -
changing
o
he
summa ions
in
an
i e a ed
double
se ies
.
i
aij
E
IR,
i,
j
E
PN
is
a
double
sequence,
when
does
he
equali y
ij
hold?
Fo
example,
i
a
ij
>
0
o
all
i,
j,
hen
his
condi ion holds
(whe e
he
sums
may
be
in ini e),
and,
in
gene al,
i
i=1j=1 j=1i=1
00 00
ITERATED
SERIES
AND
THE
HELLINGER-TOEPLITZ
THEOREM
CHARLES
SWARTZ
We
show
ha
an
i e a ed
double
se ies
condi ion
due
o An osik
implies
he
uni o m
con e gen e
o
he double
se ies
.
An
applica-
ion o
An osik's condi ion
is
gi en
e
he
de i a ion
o
a
ec o
o m
o
he
1
-
lellinge -Toepli z
Theo em
.
Tha
is,
57
1
:
l
aij
1
<
oo,
hen
he
equali y
holds
([4])
.
Ano he
condi ion
which
i-1
j-1
gua an ees
he
equali y
is
he
exis en e
o
he
double
limi ,
lim
00 00
along
wi h
he
con e gen e o
he
se ies
Eaij,
E
aij ([4])
.
The
exis-
j=1
i-1
en e
o
his
double
limi
is
o en
di iclu
o
e i y
;
one
possible
way
o
gua an ee
he exis en e
o
he
double
limi
is
o
show
ha
one
o
he
i e a ed
se ies
is
uni o mly
con e gen ,
bu
his
is
also
o en
di icul
o
e i y
.
In his
no e
we
would
like
o
poin
ou he
exis en e
o a
condi-
ion
due
o
P
.
An osik
which
in ol es
only
he
i e a ed
se ies
and which
gua an ees
he exis en e
o
he
double
limi
and,
hence,
he
equali y
o
he
wo
i e a ed
se ies
([1])
.
An osik's
condi ion
wo ks
equally
well
o
ec o - alued
se ies
so
we
p esen
his
e sion
.
To
illus a e
he
u ili y
o
An osik's
condi ion,
we
es ablish
a
Hellinge -Toepli z
ype
heo em
168

CH
.
SWARTZ
conce ning
he con inui y
o
ma ix
ans o ma ions
be ween
sequence
spaces
.
Th oughou
his
no e
G
will
deno e
an
Abelian
opological
g oup
.
Le
xij
E
G
o
i,
j
EN
.
We
assume
ha
he
se ies
>
J
xij
(>
J
xij)
con e ges
o
each
i
E
Nl
(j
E
IN)
and
seek
candi ions
which
gua an ee
he
equali y
00
00

w
00
(and
exis en e)
o
he
wo
i e a ed
se ies
>
,
>
,
xij,
L
>
,
xij
.

One
j=1
m
n
such
condi ion
is
he exis en e
o
he double
limi
li n
~-
mn
u
,
i=1
j=1
x
ij
and
called
he
double
se ies
gene a ed
by
xij)
([4])
.
We
gi e
ij
a
condi ion
which
in ol es
only
i e a ed
se ies
and
which
gua an ees
he
exis en e
o
he double
se ies
.
Since
his
condi ion
in ol es
only
he
i e a ed
se ies,
i
may
some imes
be
easie o
check
han
he
exis en e
o
he
double
limi
.
We
gi e
an example
o
such
a
si ua ion
in
p o ing
he
Hellinge -Toepli z
esul
gi en
in
Theo em
3
.
Recall
ha
a
se ies
~
xn
con e ges
in
G
o
e e y
subsequence
{ni}
.
The
p incipal
ool
i=1
used
in
he
p oo
o
ou
main
esul
on
double
se ies
is
he ec o - alued
gene aliza ion
o
subse ies
con e gen
o
he
classical
Schu
Lemma
en
weakly
con e gen
se ies
in
1
1
([2,
8
.1])
.
xi,,
con e ges
o -
cae[¿
inc easing
se-
J=I
.
quence
o posi i e in ege s
{m
j }
.

Then
he
double
se ies
Theo em
1
.
Suppose
e ges
an,d
00
00
di e ence
be ween
he wo
se ies
x
i
in
G
is
subse ies
con e gen
i
he
se ies
x
ij
=
LL
xii
.
i=1
j=1

j=1
i=1
P oo -
No e
ha
he
se ies

xij
con e ges
o
each
j
(conside
he
i=1
ce
00
xi,,,
whe e
i=1
j=1

i=1
j=1
nk
=
k
'o
each
kand
{mk}
is
he
subsequence
{1,
. . . ,
j
-
1,
j
+
1
. . . .
})
.
xin~
an
00 00
j
(deno ed
con-
IT RAT n
SERIES

169
00 00
Thus,
i
a
C
IN
is
ini e,
he
meaning
o
i=1
jEo
in ini e,
a ange
he
elemen s
o
a
in o
a
subsequence
{ni,
n2,
. . .
}
and
se
00 00

00 0o

m
xij =
1
:
Y
:xinj
.
Se
z
n
,,
j
=
57
xij
.
Then
o
a
C
N,
Y
:
Zncj
=
m
8
.1]),
he
se ies
jEo
i=1
exis s
and
equals
ij
con e ges
o
=1jE
o
i=1

jEo
as
m
-> oo
.
By
Schu 's
Lenl la
([2,
;)
is
subse ies
con e gen
and
li n
m
00 00

00 00
xij
is
clea ,
i
a
C
N! is
j=1
i=1

i=1
jEo
n ,
n
xij uni o mly
'o
a
C
IN
.
Hence,
he
double
li ni
liin
>
,
>
,
X¡33
m
'
¿
Example
2
.
The
condi ion
in
Theo em
1 is
su icien
o
he
exis ence
o
he double
;
se ies
(and
he
equali y
o
he
2
i e a ed
se ies),
bu
¡
is
no
necessa y
.
In
he
scala
case
he
hypo hesis o
Theo e l
1
iinplies
ha
he
ows
o
he
ma ix
[x ij
]
a e
absolu ely
con e gen ,
so
he
ma ix
xij
=
(-1)j+
l
/i
2
j
a¡ls
o
sa is y
he
lypo hesis
o
Tlleo cin
1,
bu
i e
;
double
se ies
5--
x
ij
con e ges
.
Z
I
.7
Theo em
1
was
p o ee
o se ies
in
a
space
;
cquippcd
wi h
a
scqucn-
ial
con e gence
s uc u e
sa is ying
ce ain
con e gence
p ope ies
by
An osik
in
[1,
3
.3]
.
An
in e es ing
aspec
o
he
hec em,
e en
o
scala, -
alued
se ies,
is
ha
he
condi ion
in
he
hypo hesis
o
he
heo e l
o lly
in ol es
i e a ed
se ies
and,
he e o e,
can
some imes
be
easily
checked
.
We
gi e
an example
o
such
a
condi ion
in
he
p oo
o
he
Helli lge-
; -
Toepli z
esul
below
.
The
classical
Helli lge -Toepli z
Theo e l
asse s
lla
a iy
ma ix
which
maps
12
¡ i o
12
is
(no m)
con inuous
([5])
;
wc
seek
condi ions
on
sequence
spaces
which
will
gua an ee
ha
ma ix
ans 'o ml io ls
be ween
he
sequence
spaces a e
con inuous
.
Since
Theo em
1 is
alid
o
ec o - alued
se ies,
we
conside
ec o - alued
sequence
spaces
.
Le
X,
Y
be
Hausdo
opological ec o
spaces
and
le
L(X
;
Y)
be
he
space
o
all
con inuous
linea
ope a o s
om
X
in o
Y
.
Le
E(F)
be
a
ec o
space
o
X- alued
(Y- alued)
sequences
;
i
x
E
E,
we
deno e
;
he
k
`
h
coo dina e
o
x by
xk
so x
=
{xk}
.
The,-dual
o
E
(wi h
espec
o
Y),
deno ed
by
E",
is
he
space
o
all
sequences
{Tk}
=TC
L(X,
Y)
such
170

C11
.
SWARTZ
ha
he
se ies
k=1
equi e ha
he
ope a o s
a e
con inuous)
;
i
Y
is
lle
scala
ield,
we
00
w i e
E
l
9y
=
Ep
.
I
x
E
E
and
T
E
Ep
y
,
we
w i e
T
-
x
=
Tkxk
.
k=1
This
gi es a
map
x
-
T
-
x om
E
in o
Y, and
we
le
u
(E,
EOY)
be
he
weakes
opology
on
E
such
ha
all
o
heses
maps
o
T
E
EQY
a e
con inuous
;
when
X
and
Y
a e
he
scala
ield,
his
is
jus
he
weak
opology
0'(E,
EA)
om
he
duali y
be ween
E
and
i s
/O-dual,
Ep
.
Le Aij
E
L(X,
Y)
o
i,
j
E
N
and
le
A
be
he
ope a o - alued
ma ix
[Aij]
.
We
say
ha
A
maps
E
in o
F
o
A
E
(E,
F)
i
o
each
x
E
E,
i
E
00
Tkxk
con e ges
o
each
.x
E
E
([9],
Maddox
does
no
IN,
he
se ies
E
Aijxj
con e ges
and
he
sequence
{E
Aijxj
}
E
F,
Le
.,
j=l

j=1
i
he
o mal
ma ix
p oduc
Ax=
{
j=1
xE
E
.
We
a e
in e es ed
in
condi ions
which
gua an ee
ha
a
ma ix
A
E
(E,
F)
is
con inuous
wi h
espec
o
app opia e
opologies
on
E
and
F
.
Fo
example,
he
classical
Hellinge -Toepli z
Theo em
asse s
ha
any
(scala )
ma ix
A
E
(l2,
12)
is
no m
con inuous
([5])
.
Toepli z
and
Kó he
gene alized
his
esul
o
o he
sequence
spaces
([8],
[7,
34
.7(7)])
.
We
now
use
Theo em
1
o
gi e
a
u he gene aliza ion
o
he
Toepli z-
Kó he
esul
o
ec o - alued
sequence
spaces
.
The
pai
(X,
Y)
is
said
o
ha e
he
Banach-S einhaus
p ope y
i
when-
e e
Tk
E
L(X,
Y)
con e ges
poin wise,
lim
Tkx
=
Tx,
o
x
E
X,
hen
he
limi
ope a o
T
is
con inuous
.
Fo
example,
i
X
is
an
F-space
o
i
X
is
a
ba elled
locally
con ex
space
and
Y
is
a
locally
con ex
space,
(X,
Y)
has
he
Banach-S einhaus
p ope y
.
The
sequence
space
E
is
said o
be
mono one
i
moE
=
E, whe e
mo
is
he
scala
sequence space
consis ing
o
all
sequences
wi h
ini e
ange
and
moE
is
he
coo dina ewise
p oduc
o
sequences
in
mo
and
sequences
in
E
([3])
.
In
pa icula ,
any
no mal
(scala )
sequence
space
is
mono one
([6,
30
.1])
.
Fu he ,
coo
(X)
deno es
he
space
o
X- alued
sequences
which
a e
0
e en ually
;
i
X
is
he
scala
ield
we
w i e
coo(X)
=
coo
.
We
now
es ablish
ou
Hellinge -Toepli z
esul ,
which
asse s
ha
a
ma ix
ans o ma ion
A
E
(E,
F)
is
con inuous
wi h
espec
o
he
weak
opologies
o,(E,
E
AY
),
Q(F,
FO
Y
)
o
E
and
F
unde
app op ia e
condi ions
on
E
and
F
.
Aijxj}
belongs
o
F
o
each
Theo em
3
.
Le
E
he
mono one
and
con ain
coo(X)
and
le
(X,
Y)
ha e
he
Banach-S einhaus
P ope y
.
I
he
ma iz
A
=
[Aij]
maps
E
in o
F,
hen
A
is
u(E,
EO
Y
)
-
o,(F,
FO
Y
)
con inuous
.
P oo
. .
Le
B=
{Bi
}
E
FP
Y and
le
Ai be
he
ieh
ow
o
A
so
B
.
Ax
=
Bi(A'
-
x)
_
BiAijxj
o
x
E
E
.
No e
o
each
j
he
se ies
j
BiAij
con e ges
in
he
s ong ope a o
opology
o
L(X,
Y)
(Fix
j
and
o
x
E
X
le
x
be
he ec o
in
E
wi h
x
in
he
jeh
coo dína e
and
0
elsewhe e
.

Then
i
:
Aikxk
=
Aijx
so
{A
ij
x}i
E
F
and
since
k
A
ij
x
con e ges,
Le
.,
E
BiAij
con e ges
in
he
s ong
i
ope a o opology
o
an
elemen
o
L(X,
Y)
since
(X,
Y)
has
he
Banach-
S einhaus P ope y)
.
Since
E
is
mono one,
he
se ies
~
J
~
B
i
A
j ,~,
xn
,
j
BEFQ
Y
,
ITERATED
SERIES

171
con e ges
o
each
xE
E
and
subsequence
{n
j }
.

By
he
In e change
Theo em
1,
i
we
se
Cj
=
i
:
BiAi
;
and
C
=
{C
j },
hen
C
E
EQ
Y
and
B-Ax=i
:
i
j

j
i
E
which
is
a
(E,
EQ
Y
)
con e gen o
0,
hen
Ax
óis
u
(F,
F
13
Y)
con e gen
o
0
and
A
is
con inuous
wi h
espec
o
hese
opologies
.
B
i
A
ij
x
j
=
EE
B
i
A
ij
x
j
=
C
-
x
so
i
{x
6 } is
a
ne in
The
(scala )
space
12
ob iously
sa is ies
he
hypo hesis o
Theo em
3
and
i
A
E
(12,12),
hen
A
is
weakly
con inuous
and,
hence,
no m
con inuous
;
his
is
jus
he
classical
Hellinge -Toepli z
Theo em
.
Mo e
gene ally,
i
E
and
F
a e
scala
sequences
and
E
is
mono one
and
con~
ains
coo,
hen any
ma iz
map
A
:
E
--->
F
is
con inuous
wi h
espec
o
he
weak
opologies
o,(E,
EQ) and
o,(F,
Fa)
.
In
pa icula ,
i
E
is
no mal
(Le
.,
i
x E
E
and
Jyk1
<
jxk1
o
all
k,
hen
y
E
E),
hen
E
is
ob iously
mono one
so
he
esul
holds
in
his
case
;
his
is
essen ially
he
e sion
o
he
Hellinge -Toepli z
Theo em
o
sequence
spaces
due
o
KS he
([7,
34
.7
.(7)])
.
Ko he's
esul
uses
a-duals
( he o¿-dual
o
a
scala
sequence
space
E
is
he
se
o
all
sequences
{yj
}
such
ha
_
jxj
yj
1
<
oo
o
all
x
E
E
;
o
mono one
spaces
E° =
EA)
so
is
so newha
weake
han
Theo em
3
since
u(F,
Fp)'is
s onge
han
u(F,
Fa)
.
In
he
scala
case
he
p oo
o
Theo em
1
abo e
uses only
he
scala
e sion
o
he
Schu
Lemma
([2,
8
.2])
and
he
p oo
o
Theo em
3
is
hen
much
mo e
elemen a y
han
ha
gi en
by KS he
which
uses
esul s
on
p ojec i e
limi
opologies
o
locally
con ex
spaces
.
A
scala
e sion
o
Theo em
3
has
been
es ablished
in [11]
.
I
Y
is
he
scala
ield,
he
spaces
E
and
EAY
=E
13
a e
in
duali y
wi h

172

Ci-1
.
SWAlUZ
espec
o
he
bilinea
pai ing
x
-
y,
whe e x
e
E
and y
is
an
X'- alued
sequence
belonging
o
EQ
([10])
.
I
E
is
mono one
and
X
is
ba elled,
he
hypo hesis
o
Theo em
3
a e
sa is ied
so
any ma ix
map
A
om
E
in o
F
is
con inuous wi h
espec
o
he
weak
opologies
u(E,
EO)
and
u(F,
FO)
.
In his case,
which
includes
he
case
when
X
is
also
he
scala
ield,
he
ma ix
mapA
o
E
in o
F
is
also
con inuous
wi h
espec
o
he
Mackey
(s ong)
opologies
o
E
and F,
espec i ely
([10])
.
Mo eo e ,
he
co npu a ion
in
Theo em
3
shows
ha
he
anspose
o
he
ope a o A,
A'
:
FQ
->
EA,
is
gi en
by
he
ma ix
[A
íi
] .
In
his case,
he
anspose
map,
A',
is
con inuous wi h
espec
o
he
weak
(Mackey,
s ong)
opologies
o
FQ
and
El
3
,
espec i ely
([7,
32
.2])
.
The e
a e
abundan
examples
o
ec o
sequence
spaces
sa is ying
he
hypo hesis o
Theo em
3
.
Fo
example,
coo(X)
o
co(X),
he ec o
space
o
all
X- alued
sequences
which
con e ge o
0,
o
mo(X),
he
ec o
space
o
a,ll
X- alued
sequences
wi h
ini o
;
ange,
o
l`w(X),
he
space
o
all
X'- alued
bounded
sequences, a e
all
mono one
sequence
:
spaces
con aining
coo(X
)
.
I
X
is
a
no med
space
a d
1
<
p
<
oc,
he
space
mono one
and
con ains
coo(X)
.
Re e ences
lp(X)
consis ing
o
all
X- alued
sequences
such
ha
E
11xkl1
p<
oo
is
k-
1
.

P
.
ANTOSIK,
"On
in e change
o
limi s,
gene alized
,nc ions,
con-
e
yen,ce
s uc u es
and
hei
applica ions,"
Plenum
P ess,
N
.Y
.,
1988,
pp
.
367-374
.
2
.
P
.
ANTOSIK
AND
C
.
SWARTZ,
"Ma iz
me hods
in
Analysis,"
Sp inge
Lec u e
No es
in
Ma liema ics
1113,
Sp inge -Ve lag, Hei-
delbe g,
1985
.
3
.

G
.
BENNI-TT,
A
new
class
o
sequence
spaces
wi h
applica ions
in
sumniabili y
l co y,
J
.
eine
angew
.
Ma h
.
266
(1974),
49-75
.
4
.

T
.J
.
Bizomwicn,
"An
in oduc ion
,o
he
heo y
o,
in ni e
se ies,"
MacMillan,
London,
1926
.
5
.

E
.
H a
.i,INCC
:
AND
O
.
TO p
.<< z,
G ündlagen
ü
Bine
Theo ie
den
une dlichen
Ma i
en,
Ma h
.
Ann
.
69
(1910),
289-330
.
6
.

G
.
Kó" uE, "Topological
ec o
spaces
I,"
Sp inge -Ve lag,
N
.Y
.,
1969
.
7
.

G
.
KÓTIm,
"Topological
ec o spaces
II,"
Sp inge -Ve lag,
N
.Y
.,
1979
.
IT aZATED
SERIES

173
8
.

G
.
KóTiIG
AND
O
.
TOEPLITZ,
Linea e
Rliu ne
mi
unendlichen
io-
len
Koo dina en
und
Ringe
unendlichen
Ma izen,
J
.
eine
angew
.
Ma h
.
17
(1934),
193-226
.
9
.

I
.
MADDOX,
"In ini o
ma ices
o
ope a o s,"
Lec u e
No es
in
Ma hema ics
786,
Sp inge -Ve lag,
Be lin,
1980
.
10
.

N
.
PHOUNG-CÁC,
Su
les
espaces
pa ai
de
sui es géné alises,
Ma h
.
Ann
.
171
(1967),
131-143
.
11
.
C
.
SWARTz,
Weak
sequen ial
comple eness
o
sequence
spaces,
p ep in
.
Depa me i
o
Ma hema ical
Sciences
College
o
A s
and
Sciences
Box
30001
Dep
.
3MB
Las C uces
NLW
MGXICO
88003-0001
P ime a
e s ió
ebuda
el
8
d'Ab il
(le
1991,
da7 e a
e sió
ebuda
el
2
de
Se eu b e
de 1991