Re .R.Acad.Cien.Exac .Fis.Na . (Esp)
Vol.
92,
N.o
1,
pp
27-33, 1998
Ma emá icas
COMPLETENESS OF SPACES WITH TOEPLITZ DECOMPOSITIONS1
(Decomposi ions/Comple eness/Su n nabiIi y/Bases)
PEDRO
J.
PAÚL,
CARMEN
SÁEZ,
JUAN
M.
VIRUÉS
Depa amen o de Ma emá ica Aplicada
JI.
Escuela Supe io
de
Ingenie os. Uni e sidad
de
Se illa.
P esen ado
po
José Bone , 3
de
diciemb e
de
1997.
Acep ado
el
14
de
ene o
de
1998.
ABSTRACT
We ex end o hese ing
o
Toepli z decomposi ionso
alocally con ex space Ein o subspaces (Ek)a esul abou
Schaude decomposi ions due o Kal on ha links he com-
ple eness
o
E o he comple eness
o
bo h he decomposi-
ion and he pieces
(E
k
).
The p oo
is
simpli ied by he
applica ion
o
adouble limi echnique. Then we s udy
wha we call he Ga ling opology
o
aspace wi h a
Toepli z decomposi ion wi h espec o ama ix Tin he
amewo k
o
he .B duali y
o
sequence spaces.
RESUMEN
En es e abajo ex endemos al ma co de las descom-
posiciones de Toepli z de
un
espacio localmen e con exo
Een subespacios
(E
k)un esul ado sob e descomposiciones
de Schaude debido aKal on que liga la comple i ud de E
con la comple i ud de la decomposición ylas de las piezas
(E
k
);
el uso de una écnica de lími e doble nos pe mi e
simpli ica la p ueba o iginal. Pos e io men e es udiamos
lo que llamamos opología de Ga ling de
un
espacio con
una descomposición de Toepli z con espec o auna ma iz
Ten
el
ma co de la
.BT-dualidad
de espacios de sucesiones.
INTRODUCTION
Up
o
wha poin can one subs i u e o dina y summa-
bili y by ama ix summabili y me hod in he de ini ion
o
aSchaude decomposi ion and, s ill, ob ain nice esul s
abou he locally con ex s uc u e
o
he space
in
e ms
o
he locally con ex s uc u e
o
i s pieces? Ou pu pose
he e
is
o ex end he cha ac e iza ion
o
he comple eness
o
aSchaude decomposi ion ob ained by Kal on (9), (10)
o
he se ing
o
decomposi ions de ined
in
e ms
o
mo e
gene al ma ix summabili y me hods. The p oo is simpli-
ied by he applica ion
o
adouble limi echnique.
Gi en an in ini e ma ix
Tone
can conside i s con e -
gence ield cTand de ine, o asequence space
A,
i s co -
esponding
.BT
-dual
APT.
This dual pai has been s udied by
Bun inas (2), (3), Meye s (12) and Noll (14) as acon in-
ua ion
o
he~-duali y
heo y
o
Ga ling (5), (6). Re e we
use he heo em abou comple eness o s udy wha we call
he Ga ling opology, because i
is
ana u al ex ension
o
he <Jy- opology de ined in (5),
o
aspace wi h aT-decom-
posi ion and, in pa icula , we gi e so ne applica ions
o
he dual pai
(A,
APT).
Te minology
and No a ion. Al hough ou no a ion
and e minology will be mos 1y s anda d, e.g.
<p
is he space
o
ini ely nonze o sequences, Cis he space
o
con e gen
sequences,
e[k]
s ands o he k- h uni sequence (we e e
he eade o (16), (17), (19) and
(21»,
le us ecall a ew
ac s om summabili y heo y.
Le
T=
Und
be an in ini e
ma ix
o
scala s om he ield K
o
eal o complex num-
be s. The ma ix T
is
said o be: ow- ini e
i
each ow
o
Tis in
<p,
an Sp¡-ma ix
i
each column
o
Tis con e gen
o
1,
and e e sible
i
o e e y sequence yEc he in ini e
sys em
o
linea equa ions T.x
==
yhas uniquesolu ion.
I
is
well-known (21, 5.4.5-5.4.9) ha each ow- ini e and
e e sible Thas aunique wo-sided in e se ma ix
TI
such
ha each ow
o
11
is
in II and o each yEc he unique
solu ion
o
T.x=yis 11•
y.
Le Ebe alocally con ex space. The con e gence ield
o
T
in
Eis he space cT(E)
o
all sequences
(x
k) am E
such ha he p oduc T.(xk)
is
acon e gen sequence in
E.
Fo (xk)Ec (E) he limi
o
he
sequence T.(xk)
is
called he T-limi
o
(x
k)and will be deno ed by T-lim .xk,
in
o he wo ds
IThis esea ch has been pa ially suppo ed
by
la Conseje ía de Educación yCiencia de
la
Jun a de Andalucía and
by
la Di ección de In es igación
Cien í ica yTécnica, p ojec PB94-l460.
28 Ma emá icas: Ped o
J.
Paú]
e
al.
Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998;
92
We simply deno e by cT he con e gence ield
o
T
in
JK.
I
T
isa
ow- ini e and e e sible ma ix hen he no m
IIxll
:=
liT·
xlL
makes CTaBanach space (isomo phic o
c).
De ini ions. Le T=
[ "d
be a ow- ini e in ini e
ma ix
o
scala s. Asequence
(P
k)
o
non- i ial, mu ually
o hogonal and con inuous linea p ojec ions de ined on a
locally con ex space E
is
said
o
be a
Toepli z
decompo-
si ion
o
Ewi h espec
o
he ma ix To , sho Iy, aT-
decomposi ion
o
E,
i
x=T
-lim
J x
o
e e y xE
E.
Al e na i ely,
i
we de ine he sequence
o
ope a o s
T"
:xEE
-7
T,,(x):=
.~,>"kPkX
E
E,
k
hen
(P
k)
is
aT-decomposi ion
o
Ewhene e lim"
T~
=
x o e e y xE
E.
I
is
impo an o no e ha he ope a o s T,,'s a e no
p ojec ions in gene al ( hey a e inc easing p ojec ions in
he case
o
aSchaude decomposi ion), howe e we do
ha e T"Pk=PkT" =
"k
Pk o all n, kEN. No e also ha
he sequence
o
ope a o s (T,,) is p ecisely he p oduc T .
(P
k)
hence saying ha lim"
T"
x=xis he same as saying
ha he sequence
T·
(P0)
con e ges o
x.
Call Ek:=
PiE).
Since Ekdoes no educes o he ze o subspace and o
e e y xkEEkwe ha e xk =lim"
T,h
=lim" ,,0k' i ollows
ha lim"
nk
=
1,
i.e., Tis an
SP ma ix.
S ill ano he way
o
lookinga
aToepli z decomposi-
ion
is
he ollowing: E e y Ekis acomplemen ed sub-
space o Eand we can iden i y e e y xEEwi h he ec-
o - alued sequence (pkx) E
I1
Ek,so ha Ebecomes a
linea subspace
o
I1
Ek ha , wi h he opology ansla ed
om
E,
has he
se
o
all ini e sequences as adense sub-
space because lim" T"x =x o e e y xEEand T
is
ow"
ini e.
AT-decomposi ion
(P
k)
o
alocally con ex space Eis
said o be:
ini e-dimensional
i
e e y Ek
is
ini e-dimen-
sional;
equicon inuous
i
he sequence
o
ope a o s
(T
n)
is
equicon inuous; and
comple e
i
o each squence
(x
k)
E
I1
Eksuch
ha
he p oduc T.
(Xk)
is
aCauchy se-
quence in E he e exis s xEEsuch ha xk=
P0
o e e y
kENand, a
o io i,
T·
(x
k)con e ges
o
x.
Example 1. E e y Schaude decomposi ion is a
Toepli z decomposi ion wi h espec o he o dina y sum-
mabili y ma ix L=
IO'"d,
whe e
0"k
=1
i
n
::;
kand 0
nk
::::
Oo he Wise.
Example 2. ACesa o basis induces aone-dimension-
al
Toepli z decomposi ion wi h espec o
el
.
L,
whe e el
=
[c,~d
is
he Cesa o ma ix
o
o de 1de ined by
C/~k
::::
n-
l
i
n
::;
kand
C],k
=oo he wise. (The ma ix el .L
is
some imes called he se ies- o-sequence Cesa o ma íx.)
Decomposi ions
o
Banach spaces wi h espec o Cesa o
ma ices
We e
i s ly conside ed by Bu ze and his collab-
o a o s in Aachen (see (19, pp. 785 and 801
o
ol.
I1)).
Example
3.
AK-space is alocally con ex sequence
space }¡,::J
cp
such ha he k- h p ojec ion de ined by
í' k((x,,)J
:=
xke[k]
is
con inuous o e e y kE
N.
A
K-
spaceA
is
said
o
ha e p ope y
T-AK
i
x=T-lim
xke[k]
o e e y sequence x=
(x
k)E
A.
Thus, asequence space
Ahas p ope y
T-AK
i
and only
i
he sequence
(J
k)is a
(one-dimensional) T-decomposi ion
o
Ao , in o he wo ds,
he sequence o ope a o s de ined
by""
:=
L../"kí' k,i.e.
(",,)
:=
T.
(í'
k
),
sa is ies x=limn
",,(Xk)
o e e y sequence
x=
(x
k)EA(see (2), (3) o (12)). (When dealing wi h
scala sequences, we shall keep he no a ions
(1
k)and
(1:
k)
h oughou he pape ). In pa icula , Ahas p ope y
L-AK
means p ecisely ha
(e
1kJ
)is aSchaude basis
o
A.
We shall be in e es ed in ma ices Tsuch ha CThas
p ope y T-AK. These ma ices we e cha ac e ized by Bun-
inas (3, Thms. 8-10).
Bun inas's Theo em.
Le
T
be
ow- ini e
and
e e s-
ible SPI-ma ix.
Then
he
ollowing
condi ions
a e
equi a-
len :
(1) The
sequence
o
coo dina e
p ojec ions
(J
k)
is
a
T-decomposi ion
o
cp
(2)
The
sequence
o
ope a o s
(
n)
is
equicon inuous
on
cp
(3)
J
we
deno e
Ti
by[ /~i]
hen
sup
{~I~ mk "k ijII
:
m,
nE
N}
<oo.
(4)
The
dual
(c
T
)'
can
be
iden i ied
wi h
he
mul ipli-
e
space
(cT
-7
cT)
o med
by
he
sequences
y
such
ha
he
coo dina ewise
p oduc
xy
is
in
cT o
e e y
xEcT
and,
in
his
case,
he
bilinea
o m
o
he
dual
pai
is
gi en
by
The i s non- i ial examples
o
ma ices Tsuch ha
cThas p ope y
T-AK
a e he se ies- o-sequence Cesa o
ma ices
o
o de a
~
O;
his was p o ed by Zelle (22).
The e o e, o a oid c1umsy epe i ions, a
ow- ini eand
e e sible
Sp¡-ma ix
Tsuch ha cThas p ope y
T-AK
will
be called a
Zelle -Bun inas
ma ix.
Fo such ama ix Twe
de ine
b(T)
:=
sup"
11""
11,
whe e
11",,11
is
he no m
o
1:
nas a
bounded ope a o om he Banach space cTin o i sel .
No e also ha
i
T
is
aZelle -Bun inas ma ix hen cT
is
a
sum
space
in he sense
o
Ruckle (17).
Example 4. Le Qbe an open, bounded and balanced
subse
o
cm
and le
A(Q)
be he space
o
all unc ions ha
Ma emá icas: Ped o
J.
Paúl e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998;
92
29
a e holomo phic on Qand can be ex ended con inuosly o
he c10su e
o
Qendowed wi h he opology
o
uni o m
con e gence on
Q.
I
is
men ioned wi hou p oo in (15)
ha A(Q) has he bounded app oxima ion p ope y. As a
ma e
o
ac , wha happens is ha
A(Q)
has an equicon-
inuous and a ini e-dimensional Toepli z decomposi ion
wi h espec o he se ies- o-sequence Cesa o ma ix Cl.
L. This decomposi ion
is
he na u al one gi en by he
Taylo se ies: Each E
A(Q)
can be uniquely w i en as
O
=
L~=OPk(J)O
(poin wise con e gence), whe e each
Pi )
is ak-homogeneous polynomia1. Now, he se
o
aH
homogeneous polynomials is dense in
A(Q)
( o ap oo
see (1), he me hod u ilized in Sec ion 1
o
ha pape can
easi1y be adap ed o show he p esen esul ). On he
o he
hand, he co esponding sequence
o
ope a o s
(T
n)is equi-
con inuous
by
(1, Lemma 1.1) o (13, 5.2 P oposi ion),
whe e i
is
shown ha
ITII(J)(z)1
~
11
o
aH
zE
Q.
Finally, as anda d a gumen abou equicon inuous se s (11,
§39.4.(1) shows ha =T-lim
Pi )
o all EA(Q).
COMPLETENESS
OF
SPACES
WITH
TOEPLITZ
DECOMPOSITIONS
Ou i s pu pose is
o
ex end o he se ing
o
Toepli z
decomposi ions a esul due o Kal on (10) ha links he
comple eness
o
E o he comple eness
o
bo h he decom-
posi ion and he pieces Ek.We shall make use
o
adouble
limi echnique ha lies behind he p oo gi en by Kal on
o Schaude decomposi ions. The Double Limi Lemma is
ce ainly well-known o double sequences bu
we
need a
e o mula ion in e ms
o
adouble ne ha can be p o en
analogously.
Double
Limi
Lemma.
Le
Ebe alocally con ex
space
and
{xij :
(i,
j)
EJxJ}
be
adouble
ne
in Esuch
ha
o
each iEJ he e exis s he limi
y¡
=limj
xij
and
o
each jEJ he e exis s he limi Zj =limi
Xii'
J
he con e -
gence
o
(xij)j o
y¡
is uni o m in J hen he h ee ne s (Xi)'
(y)
and
(z)
a e Cauchy ne s. I , in addi ion, Eis comple e
hen he h ee
ne s
aboye a e con e gen o he same
limi o
Theo em
1.
Le
(P
k)
be
an
equicon inuous T-decom-
posi ion
o a
locally con ex space
E.
Then he ollowing
a e
equi alen :
(1) Eis comple e ( esp. quasi-comple e
o
sequen-
ially comple e).
(2)
(P
k)is comple e
and
each Ekis comple e ( esp.
quasi-comple e
o
sequen ially comple e).
P oo!
I
is
c1ea
ha (1) implies (2), so
we
ha e o
show ha (2) implies (1). We shall deal only wi h he
comple eness case because he p oo s o he
h eecases
a e essen ially he same. Le
(z);,=/
be aCauchy ne in
E.
Fo each kEN he e exis s xkEEksuch ha
(P
k
Z)¡E/
con e ges
o
xk
because Pk
is
con inuous and Ek
is
com-
ple e. Since Tis ow- ini e, o e e y nENwe ha e
On he o he hand, limnT
nZ¡
=
Z¡
o
e e y iEI. To see
ha we can apply he Double Limi Lemma o he double .
ne {Tnzi:
(i,
n) EJx
N}
,le
us
check ha he con e -
gence
o
(TnZ)iE/
is
uni o m in N. Gi en acon inuous sem-
ino m
q/
on
E,
he e exis s acon inuous semino m
q2
such
ha
because
(P
k)is an equicon inuous T-decomposi ion. Since
(Z)¡E/ is aCauchy ne , i ollows ha he e exis s so ne
index ioEJsuch ha
q2(Zi
~
z)
::;
1whene e
i,
j
~
io'
The e o e,
ql(TnZ¡
-
Tnz
j)
~
1 o all nENand
i,
j
~
io.
Take limi s in j o ob ain
This shows ha he con e gence
o
(TnZ)¡E/
is uni o m
in N. The Double Limi Lemma ells
us
ha he p oduc
T·
(x
k)=
(Lk llkXk)
is aCauchy sequence. Since
(P
k)
/leN
is
acomple e T-decomposi ion, he e exis s XEEsuch ha
xk=P
~
o e e y
kE
Nand T.(xk)con e ges o
X.
Finally, since he ne (Z)¡E/ is con e gen in he comple ion
o
E, he Double Limi Lemma ells
us
now ha (z¡hE/ mus
con e ge o x
as
well
.•
Co olla y
1.
Le
(P
k)
be
an
equicon inuous
and
i-
ni e-dimensional Toepli z decomposi ion
o
alocally con-
ex
space
E.
Then he ollowing
a e
equi alen :
(1) Eis comple e.
(2) Eis quasi-comple e.
(3) Eis sequen ially comple e.
(4) .
(P
k)is comple e.
Co olla y
2.
J
a
ba elled
and
sequen ially comple e
locally con ex space has a ini e-dimensional Toepli z de-
composi ion hen
i
is comple e.
Rema k.
An
ex ended
Schaude
basis
o
alocally
con ex space Eis a amily (X)iE/ wi h he p ope y ha o
e e y xEE he e is aunique amily (a;(x))¡E/
o
scala s
such ha xcan be
w i enas
x=
L¡a¡(x)x¡
and he
unc ionals x
~
a¡{x) a e con inuous. Webb (20) p o ed
ha asepa able, non-comple e, Mon el locally con ex
space canno ha e any ex ended Schaude basis; ou Co -
olla y 2shows ha i canno ha e any ini e-dimensional
Toepli z decomposi ion nei he .
THE
GARLING
TOPOLOGY
OF
A
SPACE
WITH
A
TOEPLITZ
DECOMPOSITION
Le Ebe alocally con ex space wi h aT-decomposi-
ion
(P
k
).
In his sec ion we will see ha he e exis s a
30 Ma emá icas: Ped o
J.
Paúl e
al.
Re .R.Acad.Cienc.Exac .Fis.Na .
(Esp),
1998; 92
coa ses E'-pola opology on E o which
(P
k)is
anequi-
con inuous Toepli z decomposi ion. This opology u ns ou
o be ana u al gene aliza ion
o
he ay- opology in o-
duced
by
Ga ling in his deep s udy
o
he
~-duali y
be-
ween sequence spaces (5), (6) and, acco dingly, will be
called he e he Ga ling opology
o
E.
In he case
o
Schaude decomposi ions his opology has been s udied
by Kal on (9).
Using p imes o deno e adjoin ope a o s, o e e y xE
Eand e e y uE
E'
we can w i e
(x,
u)
=
lim(T"x,
u)
=lim
L, "k(llx,
u)
=
1I
.
II
.k
This shows ha
(p~)
is
also aT-decomposi ion
o
El
endowed wi h he weak opology a E',
E).
I
we call Ek
:=
PiE)
and
E~
:=
P~(E')
hen he dual
o
Ekcan be iden-
i ied wi h
E~.
The compu a ion aboye also shows ha he
sequence
(T;,u)
is a(E', E)-bounded.
De ini ion. Le (Pk)
be
aT-decomposi ion
o
alocal-
ly con ex space
E.
The Ga ling opology
o
Eis he pola
opology
yy{E,
El)
o
uni o m con e gence on he amily
{(T;,u) :uE
E}.
Al e na i ely,
yy{E,
El) is gene a ed by
he amily
o
semino ms
xEE
~
supl(T"x,
u)l,
(u
EE).
"
The Ga ling and he weak opology coincide on each
Ekbecause o all xkEEkand uEEl
we
ha e
we
shall make use
o
his ac acouple
o
imes.
The p ope ies
o
he Ga ling opology depend hea ily
on
he T-AK p ope y
o
he con e gence ield associa ed
o he ma ix
T.
No e ha his
is
gi en o ee in he case
o
aSchaude decomposi ion:
(e[k
l)is aSchaude basis
o
he space cs
(=
cL)
o
all summable sequences. To see how
o connec he Ga ling opology wi h he p ope ies
o
c1'
le Fbe he ec o - alued sequence space de ined by
As we no ed aboye, Ecan be iden i ied wi h asub-
space
o
F. No e ha , using he e minology gi en in he
p e ious sec ion, Eequals F
i
and only
i
(P
k)
is
acom-
ple e T-decomposi ion
o
E[a
(E,
E')]; in his case, (Pk)
is
said o be
~ comple e
by analogy wi h he Schaude
de"
composi ion case (lO).
Fo each uE
E'
we de ine he ope a o
I
is easy o see ha /::""sa is ies he ollowing p ope "
ies
(i) Fo e e y nENand
(Xk)
EFwe ha e ha he
sequence (T"xkh is also in Fand
/::""(T,,Xk)k
=
'l"1l/::",,(X
k)k'
In pa icula ,
/::""(T,,PkX)k
=
'l"1l/::",,(P
k
X)k
o each xE
E.
(ii)
J
IT .
(x
k
)]"
s ands o he n- h elemen
o
he
sequence T.(xk
),
hen
and, in pa icula ,
11/::",,(Pkx)IIT
=
sup"
I(T"x,
u)1
o each xE
Eso ha he Ga ling opology is gene a ed by he amily
o
semino ms x
~
11/::,."
(Pkx)IIT
as
uE
E'.
P oposi ion
1.
Le T
be
aZelle -Bun inas ma ix and
(P
k)
be
aT-decomposi ion
o
alocally con ex
space
E.
Then
yy{E
,E')
is
he
coa ses El-pola opology
such
ha
(P
k)
is
an equicon inuous T-decomposi ion
o
E.
P oo!
Using ha he p ojec ions (Pk)a e weakly
con inuous on
E,
ha he Ga ling opology is s onge han
he weak opology and ha he Ga ling opology induces
on each subspace Eki s own weak opology
0"(
Ek,
E~),
i
ollows ha he p ojec ions
(P
k)a e con inuous on E o
he Ga ling opology. We now show ha (Pk)is an equi-
con inuous T-decomposi ion
o
Eendowed wi h
yy{E,
E').
Fo all xE
E,
uEE', and mENwe ha e, using
(i)
and
(ii) aboye,
11/::,."(IlT,,,x)kIIT
=
11/::,."(T,,,! X)kIIT
=
=
11'l"IA,(llx)kIIT
:,;
b(T)II/::,.,,(llx)IIT
so ha
(T
n)is aY equicon inuous sequence. (A ema k
is
in o de he e: al hough
yy{E,
El)
is
gene a ed by he amily
o semino s
sUPn
I(T"x,
u)l,
since he
T;s
a e no inc easing
p ojec ions
-as
i
is
he case
o
aSchaude decomposi-
ion-
we canno conclude di ec ly ha
sUPIII(T"Tmx,
u)1
:,;
:,;
sup"I(T"x,
u)I·)
The ollowing compu a ion shows ha x=
yy{E,
E')-
limnT
II
x o all xE
E:
whe e, in he la e s ep, we ha e used ha cThas p ope y
T-AK. Now, le be an E'-pola opology such ha (Pk)
is
an equicon inuous T-decomposi ion
o
Eendowed wi h
.
Ma emá icas: Ped o
J.
Paú1
e
al.
Re .R.Acad.Cienc.Exac .Fis.Na .
(Esp),
1998;
92
31
Fix uEE', hen he e is a -equicon inuous se De
E'
such ha
sup
1(T"x,
u)1
~
sup
I{x,
)l·
II
.
eD
This shows ha y.JE, E') is coa se han . •
Example 5.
I
Tis aZelle -Bun inas ma ix hen he
Ga ling opology on CTcoincides wi h he
IHIT
- opology.
To see his, conside he squence
e=
(1,
1,
...)E
(CT)'.
Then
o e e y x=
(X
k)EcT,we ha e
This shows ha he no n opology is coa se han he
Ga ling opology. The con e se ollows om P oposi ion
1.
(Fo he case
o
he Cesa o se ies o sequence summu-
bili y ma ix el. 2, his ac was p o ed by Flo encio (4),
using di e en echniques.)
The dual
o
E[ T(E,
E')]
can be bigge han E' (see
Rema k 2below),
bu
we can cha ac e ize i in he ollow-
ing way: Gi en uE
E'
and aE(cT
)'
we may de ine alinea
unc ional au on Eby
I
Tis aZelle -Bun inas ma ix, so ha (cT
)'
is also a
sequence space and we w i e a=(ak) hen we ha e
We
deno e by (cT
)'
•
E'
he space
o
all linea unc ion-
als hus ob ained.
P oposi ion 2. Le Tbe aZelle -Bun inas ma ix
and
le
(P
k)be aT-decomposi ion
01
alocally con ex space
E.
Then he dual space
01
Eendowed wi h i s Ga ling opol-
ogy is (CT)'· E'. In pa icula ,
yy(E,
E') is compa ible wi h
he dual pai
i
and
only
i
E'
=(cT
)'
.E'.
P oo! Gi en uE
E'
and aE
(CT)'
we ha e
The e o e, au is
YT
(E,
E')-con inuous. Con e sely, le
Zbe a
YT
(E, E')-con inuous linea unc ional on
E.
Then
he e exis s uE
E'
such
ha !(x,
z)1
~
IIA
II
(Pkx)IIT
o all x
E
E.
Iden i y Ewi h i s image in F ia he injec ion x
-4
(P
0).
This enables us o de ine alinea unc ional aby
a:
((Pkx,
u))
EA/I(E)
-4
(((Pkx, u)), a)
:=
(x,
z)
which, obiously, is well-de ined and!HIT -con inuous. By
using ha he subspaces
(E
k)a e non- i ial, i is easy o
see ha
qJ
e
A"
(E)
so ha his is adense subspace
o
cT
Finally, ex end a o all
o
cTby con inui y o ob ain
and he
p oo
is
inished. •
Rema ks. (1)
I
BTs ands o he uni ball
o
(cT) "
he equali y
E'
=
(CT)'
.
E'
is equi ulen o
E'
=BT.E', and
i
his equali y holds hen
E'
is said o be B in a ian , as
in he sequence space case (5), (9).
(2)
I
E'
is BT-in a ian hen he Ga ling opology is
compa ible wi h he dual
pai
(E,
E') and so he sequence
o
unc ionals
(uTn)n
is
.~
(E', E)-bounded in E', in which
case he T-decomposi ion is said o be simple. As he e a e
non-simple Schaude basis (see he ema ks ollowing De .
2.3 in
(9»,
i ollows ha no all Ga ling opologies a e
compa ible.
(3) I is easy o see ha aToepli z decomposi ion is
simple
i
and only
i
he
weak
and he Ga ling opologies
ha e he same amily
o
bounded se s.
We s udy now when is EIYT(E, E')] acomple e space.
P oposi ion 3. Le Tbe aZelle -Bun inas ma ix and
le
(P
k)be aT-decomposi ion
01
alocally con ex space
E.
Then E[yy(E, E')] is comple e ( esp. quasi-comple e
o
sequen ially comple e)
i
and
only
i
(P
k)is 3 comple e
and each Ek[
0"(
Ek,
E~)]
is comple e ( esp. quasi-comple e
o
sequen ially comple e).
P oo! As we poin ed ou abo e, he Ga ling and he
weak opology coincide on each
El<'
Hence,acco ding
o
Theo em
1,
we ha e o p o e ha (PJis acomple e
T-
decomposi ion
o
E[yy(E, E')]
i
and only
i
i is acomple e
T-decomposi ion
o
Eendowed wi h i s weak opology;
Le.,
~T"comple e.
The «i » pa ollows easily om he ac
ha he weak opology is coa se han he Ga ling opolo"
gy.
So, assume ha
(P
k)is acomple e
T-decomposi ion
o
E[YT
(E,
E')] and le
(x
k)E
TI
Ekbe such ha
T·
(x
k)is a
weakly-Cauchy sequence in
E,
i su ices o show ha T .
(x
k)is also a
YT
(E,
E')-Cauchy sequence. Deno e by
zn
he
n- h elemen
o
T.
(x
k), ha is
Z/I
=
L.k
nkxk'
I
is clea ha
Pkz/I
=
/lkXk
o all
n,
kENso, using (i) abo e, we
ha e
Finally, using he Y con inuous semino ns as gi en in
(ii), we ob ain
and his la e exp esion goes o ze o as
m,
n
-4
00
because
A"
(xk)is in cTand his
space
has p ope y T-AK. •

32 Ma emá icas: Ped o
J.
Paúl e al.
Re .R.Acad.
Cienc.
Exac .
Fis.
Na .
(Esp),
1998; 92
'Co oIla y.
Le T
be
aZelle -Bun inas ma ix and le
(Pk)
be
aT-decomposi ion
o
aBanach space
E.
Then iE,
E')
is
acomple e opology ( esp. quasi-comple e)
i
and
only
i
(P
k)
is
3 comple e and each Ek
is
ini e-dimension-
al ( esp. e lexi e).
Example
6. Le Tbe aZelle -Bun inas ma ix and E
be alocally con ex space wi h an equicon inuous T-de-
composi ion
(P
k
).
I
s ands o he opology
o
E,
hen
P oposi ion I ells us ha
a(E,
E')
:;;
T(E, E')
:;;
.
Example 5shows ha o E=cT he i s inequali y is
s ic , bu he second is an equali y.
On he o he hand,
i
Eis an in ini e dimensional Ba-
nach space, hen he space co(E) o med by he null se-
quences in Eis aBanach space wi h ana u al in ini e-
dimensional Schaude decomposi ion (7) and i s Ga ling
opology, which canno
be
comple e, does no coincide
wi h i s no m opology so ha bo h inequali ies a e s ic .
APPLICATION
TO
THE
~T-DUALITY
OF
SEQUENCE
SPACES
In wha ollows, Tis aZelle -Bun inas ma ix and A
s ands o asequence space con aining
<p.
The
~T-dual
o
Ais he space
A >T
o
aH
sequences ysuch ha he coo di-
na ewise p oduc xy is T-con e gen o e e y xE
A.
I
Ais aK-space wi h p ope y T-AK hen i is cIea
ha
A'
eA
P
.
I ,
in addi ion, Ais sequen ially ba elled
hen
A'
=
?J3
(see (2), (3),
(12)
and (14)). On he o he
hand,
i
Ahas p ope y T-AK hen Ae
(A'
and, by
P oposi ion 3, he equali y holds
i
and only
i
A[ T(A,
A')]
is sequen ially comple e.
Assume now ha no opology is de ined ap io i on
A.
The na u al bilinea o m
(x,
y)
~
T-lim
XkYk
malees
(A,
A >T)
asepa a ed dual pai and, cIea ly, bo h A[a(A' A
P
)]
and
AP [a(A
P
,
A)]
ha e p ope y
T-AK.
We may ask
i
he e is as onge opology on A
ha in¡
p ope y T-AK and
s ilI compa ible wi h he duali y
(A,
A
).
We shall cha ac"
e ize his opology by ex ending and combining esu1 s
gi en by Ga ling
(5)
and Schae e (18) o he
~-duali y
( he duali y de ined in e ms
o
o dina y summabili y).
Lemma.
Le Tbe aZelle -Bun inas ma ix and A
be
a
sequence
space
con aining
([J.
Then he space
A/
3
[ T(A
P
,
A)]
is comple e
and o
ase
Ce
A
P
he ol-
lowing condi ions a e equi alen
(1) C
is
T(A
P
,A)- ela i ely
compac o
(2)
Cis
T(A
P
,A)-bounded and he con e gence
o
he sequence
(' ny)
o
y
in
he Ga ling opology T(A
P
,
A)
is
uni o m wi h espec
o
yE
C.
(3) eis
T
(A
P
,
A)
-bounded and o e e y x€A he
con e gence
o he
sequence lY)
o
xy in he
¡HIT- opology
is
uni o m wi h espec
o
yE
C.
P oo!
Acco ding
oP oposi ion
3, o p o e
ha
A
P
[
Y
(A
P
,
A)]
is comple e we ha e o show ha
(1
k)is a
comple e T-decomposi ion
o
APT a(A
P
,
A)].
bu his ol-
lows om he e y de ini ion
o
PT-duaI.
By P oposi ion
1,
(~)
is an equicon inuous T-decompo-
si ion
o
AP [ T(A
P
,
A)j;
i.e.,
(
n)is a T(A
P
,A)-equicon-
inuous sequence
o
ope a o s ha con e ges poin wise o
he iden i y on A
P
hence, by using (11, §39.4(1)), we ha e
ha
('en)
con e ges uni o mly on T(A
P
,A)-compac se s.
Since
ll
(A
P
)is ini e-dimensional o e e y nE
N,
i
ollows ha o e e y nE
N,
he se
ll
(
C)
is ela i ely
compac p o ided ha Cis T(A
P
,A)-bounded. Then Ma-
zu 's Theo em (8, Thm. 1) implies
ha condi ions
(1)
and (2) a eequi alen . The equi alence
o
(2) and (3) is
cIea . •
Theo em
2. Le Tbe aZelle -Bun inas ma ix and A
be
asequen
ce
space con aining
<p.
Then he s onge
o-
pology
on
A ha has p ope y T-AK and
is
compa ible
wi h he duali y
(A,
A
P
)
is
he opology
kT(A,
A
P
)
o
uni o m con e gence on he absolu ely con ex
and
T~AP ,
A)
"compac subse s
o
A
P
.
P oo! Tha he dual
o
A[kT(A, A
P
)]
equals
?J3
ol-
lows om he Mackey-A ens's Theo em and he ac ha
he Ga ling opology is s onge han he weak opology.
Tha AIkT(A, A
PT
)]
has p ope y T-AK ollows om he
p e ious
lemma
by
simply
no ing
ha
(x
~
,,(x),
Y)
=
(x,
y
~
,,(y)) o all xE
A,
yEA
PT
and
nE
N.
Finally,
i
CeA
P
is an absolu ely con ex and
a(
A
P
,
A)
-compac
se
such ha
[im
sup
i(x
~
,,(x),
y)1
=O
o
all xE
A,
Il
)'EC
hen, again by he lemma, Cis
Y
(A
PT
,
A)
-compac o This
implies ha kT(
A,
A
P
)is he s onge opology sa is ying
he desi ed p ope ies
.•
Co oIla y
1.
Le Abe asequence space con aining
([J.
Then he s onge opology
on
A ha has p ope y
AK
and
is
compa ible wi h he duali y
(A,
}j)
is
he opology
kT
(A,
A »
o
uni o m con e gence on he absolu ely con ex
and aJ{A
J
,A)-compac subse s
o
}jo
This co olla y can also be ob ained by combining (5,
P op. 11) wi h (18, ema ks ollowing P op. 4).
Co oIla y
2. Le Tbe aZelle -Bun inas ma ix and A
be
asequence space con aining
([J.
Then
A[13(A,
A
P
)]
has
Ma emá icas:
Ped o
J.
Paúl e al. Re .R.Acad.Cienc.Exac .Fis.Na . (Esp), 1998;
92
33
p ope y
T-AK
i
and
only
i e e y
O'(.~, lT,
}..)-bounded
se
is Y (}../i ,}.. )- ela i ely compac o
P oo!
Use he heo em plus he ae ha
i
A,
is
aK-
spaee ha ing p ope y
T-AK
hen i s opological dual is
eon ained in i s
~T-dual.
Acknowledgemen s. We hank K.-D. Bie s ed (Pad-
e bo n, Ge many),
J.
Bone (Valencia, Spain),
J.
C.
Díaz
Alcaide (Có doba, Spain) and
K-G.
G osse-E dmann
(Hagen, Ge many) o hei aluable commen s and e-
ma ks.
REFERENCES
1. Bie s ed , K.D., Bone ,
J.
&Galbis,
A.
(1993), Weigh ed spac-
es
o
holomo phic unc ions on balanced domains. Michigan
Ma h. J.
40,271-297.
2. Bun inas, M. (1973).
On
sec ionally dense summabili y ields,
Ma h,
Z.
132, 141-149.
3. Bun inas, M. (1975),
On
Toepli zsec ions
in
sequence spaces,
Ma h. P oc. Camb idge Phi/os.
Soco
78,451-460.
4. Flo encio,
M.
(1981), Sob e C-dualidad yespacios C"pe ec-
os, Re .
R.
Acad. Cienc. Exac .
Fís.
Na.,
Esp.
75, 1221-1234.
(Spanish).
5. Ga ling, D.J.H. (1967), The
~-and
y-duali y o sequence spac-
es. Ma h. P oc. Camb idge Phi/os.
Soco
63, 963-981.
6. Ga líng, D.J.H. (1967), On opological sequence spaces, Ma h.
P oc. Camb idge Philos.
Soco
63,997-1019.
7. Gami , H.G., De Wilde,
M.
&Schme s,
J.
(1968), Analyse
onc ionelle,
1,
Ma hema ische Reihe, band 36, Bi khause
Ve lag, Basel.
8. Goes, S. &Welland,
R.
(1970), Compac ness c i e ia o K5 he
spaces, Ma h. Ann. 188, 251-269.
9.
Kal on, N.J. (1970), Schaude decomposi ions
o
locally con-
ex spaces, Ma h. P oc. Camb idge Phi/os.
Soco
68, 377-392.
10. Kal on, N.J. (1970), Schaude decomposi ions and comple e-
ness, Bull. London Ma h.
Soco
2, 34-36.
11. K5 he, G. (1969, 1979), Topological Vec o Spaces
1,
n,
Sp inge -Ve lag, Be lin, Heidelbe g, New Yo k.
12. Meye s, G. (1974), Gn Toepli z sec ions in FK-spaces, S udia
Ma h 51, 23-33.
13. Mujica,
J.
(1991), Linea iza ion
o
bounded holomo phic map-
pings on Banach spaces, T ans, Ame . Ma h.
Soco
324, 867-
887.
14. NolI,
D.
(1990), Toepli z sec ions and he Wilansky p ope y,
Analysis 10, 27-43.
15. Pelczy íski,
A.
(1977), Banach Spaces
01
Analy ic Func ions
and Absolu ely Summing Ope a o s. CBMS Regional Con e -
enCe
Se ies, ol. 30, Ame ican Ma hema ical Socie y, P o i-
dence,
R.
I.
16. Pé ez Ca e as,
P.
&Bone ,
J.
(1987), Ba elled Locally Con-
ex Spaces, Ma hema ics S udies, ol. 131, No h-Holland,
Ams e dam, New Yo k, Ox o d, Tokyo.
17. Ruckle, W.H. (1981), Sequence Spaces, Resea ch No es in
Ma h., ol. 49, Pi man, London.
18. Schae e , H.H. (1970), Sequence spaces wi h agi en
K5 he~
dual, Ma h.Ann. 189, 235-241.
19. Singe , I. (1970, 1981), Bases in Banach spaces
1,
n,
Sp inge -
Ve lag, Be lín, Heidelbe g, New Yo k.
20. Webb, J.H. (1974), Ex ended Schaude decomposi ions
o
10-
cally con ex spaces, Glasgow Ma h. J. 15, 166-171.
21. Wilansky,
A.
(1984), Summabili y h ough Func ional Analy-
sis, Ma hema ics S udies, ol. 85, No h-Holland, Ams e dam,
New Yo kd and Ox o d.
22. Zelle ,
K.
(1953), App oxima ion in Wi k elde n on Sum-
mie ungs e ah en. A ch. Ma h. (Basel) 4, 425-431.
SERIE «LIBROS ANTIGUOS»
REAL ACADEMIA DE CIENCIAS
kIBELLVS
YSAGOGICVS
ABD
VI:
GLORIOSI,DE'I:QVI
DIO
ADMAGISTERIVM
IVDITJO
INTERPRETATVSAIOANN
HTVMQVEIN EVNDEMAI.
EDITV~
VTlLISERlECON
..
",',
-/
Alchabi ius
Libellus isagogicus
ad
magis e ium iudicio um as o um (la ine), Johanne
Hispalensi in e p e e, cum commen o Johannis de Saxonia.
Vene iis: E ha dus Ra dol , 1485.