Pub
.
Ma
.
UAB
Vol
.
30
n2
2-3
Des
.
1986
INFRATOPOLOGICAL
BORNOLOGIES
AND
MACKEY-CONVERGENCE
OF
SERIES
MiguelA
.
Canela
and
Me cé
Se ahima
SYNOPSIS
:
In
his
pape
some
aspec s
o ín a opological
bo nologies
a e discussed
.
Fi s ,
i
is
shown
ha
bo nologies
wi h
coun able
basi's
a e
in a opological
.
Second,
i
is
shown
ha
he
con e gence
o se ies
p esen s
some
:
pa hologies
beyond
his
class
.
We
,
bllow
closely
he
,
e minology
o
[51
and [4]
,
whe e
he
main
opics
conce ning
bo nological
spaces
can
be
,
ound
.
Roughly
speaking
;,
a
bo nology
is
iü a opological
when
:can
be
de ined
om
a
opologyin
he
usual
way
:
he
bounded
se s
:
a e
hose
which
a e
abso bed
by ze o
neighbou hoods
.
Mo e
p ecisely

le
(E,
S
)
be a
linea
bo nological
space, and
TS
he
Mackey
;
closu e
opolo-
gy
associá ed
o
his
space
.
.
The Von
Neumann
bo nology
o
TS
,
de-
no ed
by
,
BIT
S
,
is
de ined
as
ollows
:
a
subse
B
C
E'
is
bounded
when
o
e e y
ze o
neighbou hood
V,
he e
is
some
A
>-0
such ha
B
C

X
V

(T
;
is
abso bed
by
V)'
.
The
bo nology
1 is
said
o
be
in-
a opolog'cal
when

0=
BT0,

_
A
mo e use ul
cha ac e iza ion,
whichdoessno
in ol e
any
opology
is
he
ollowing~
12-1
i
P oposi iom
1
.
.
A
linea
bo nological
space
(
E,
0

)

is
-
in a opo-
logical
i 'and
only i
e e y
subse
B
C
E,
which
is
abso bed
o
e e y
bo ni
.Vo ous
subse
o
&
.
is
bounded
in

(E,R)
(
a
subse
UC
E

is
bo ni o ous
when
U
abso bs e e y
bounded
se
o
(E,
S
))
.
The
boundedsubse s
o
an
o dina y
opological
ec o
space
( he
Von
Neumann
bo nology) o m an in a opological
bo nology
.
Ne e heless,bo nologies
which
a e
no
in a opological
can
appea
when we
deal
wi h
equicon inuous
se s
.
We
e e
o
(1
;
_
.
,
example
p
.
166,
o
an
equicon inuous
bo nology
no
Kolmogo o
and
consequen ly
no
in a opological
.
We
see
nex
ha
spaces
wi h
coun able
basis
a a
also
in a opo-
logical
.
P oposi ion
2
.
_Le _
(E,
S
)
be a
sepa a ed
con ex
bo nological
space
and
suppose
ha
S
admi s
a
coun able
basis

(Bn
)
n
>
1
.
Then
Bis
in-
a opological
.
P oo
.
We
can
supposse
ha
he B
n
's
a e
absolu ely
con ex
and
ha
he
sequence
is
inc easing
.
Thus,
i
we deno e
by
~~
.
~~
n
he
gauge
o
B
n
,
we ob ainan inc easingsequenceo
no ms,
each
~~
.
lin
de ined
in
E
n =
Span
B
n
.To
make i
easie ,
we
de ine
llxli n
o
x
E
E
En
.
Le
A
be an
unbounded
subse
o
(E,
S
)
.
hen,
o
e e y
n,
we
can
ind
xn E A
wi h
llxn 11 n > n2
.
Fo
e e y
k,
we de ine
:
and
pu ing
:
ñk
=
1
min
(lixlll
k
',
. .
.,
llxkllk
'
l)
.
k
V
=
U

ñ
k
.
B
k
,
k
11
we
ob ain
a
bo ni o ous
subse o
E,
and
V
does
no abso b
A,
be-
cause, o
E> 0
a bi a y,i
n
>
E
:-l,
xn
does
no
belong
o
any
AkBk
:
a)

i
k
2
n,
we ha e

11
E
xnllk
1
Ek
ñ
k
a E
n
A
k > Xk
b)

i

1

_ k

<

n,

we

ha e

11

Ex
n1 ik
?

11

Ex
n
l I
n
>
en
2

>

n
>X
k*
ApplyingP oposi ion
1,
we
conclude
ha
(E,
S
)
is
in a opo-
logical
.
People
which
is
used
o
linea
bo nological
spaces
knows
ha
such
a
space
can
ail
o
ha e
he
p ope ies
ha
one
hopes
o
ind
in
Func ional
Analysis,
unless
some
ex a
assump ions
a é
aken
.
Res ic ion
o
in a opological
spaces
is en
example
o
ex a
assump-
ion
unde which
some
pa ologies
do
no appea
.
We will
see
he e
ha
he
con e gence
o se ies
does
no
wo k in
he
usual
way
when
we
pla-
ce
ou sel es
ou side
he
class
o
in a opological
bo nologies
.
A
se ies

2
:

x
nis
said
o be
Mackey-con e gen
in
a
linea
n
>
1
bo nological
space
when
he
sequence
,
(
Y-

x
k)
n>
1
o
pa ial
k=1
sums
is
Mackey-con e gen
."The
ollowing
esul s
a e
well-known
o
he
opological
con e gence
in
a
opological
ec o
space
.
Ou
con-
e gence
has
been
conside ed
in
[6]
.
P oposi ion
3
.
(
i
)
Le
(E,
S
)
be
a
Mackey-comple e
con exbo -
nological
space,
(xn
)
n Z 1 a
bounded
sequence
in
(E
B
),
and
(an
)
n

>
1

a
sequence
o eal
numbe s
such
ha

lan
1

is
i-
n
>_
1
ni e
.
Then,

2
:
an
xn is
Mackey-con e gen
.
n
i
1
(ii)
Le
(E,
S
)
be an
in a opological
con ex
-
bo nological
space,
and
(xn
)
n
>
1
'
a
sequencesuch
ha ,
o
e e y
sequence
o
eal
numbe s
(a
n
)
n
>

1
wi h

la
n
1

ini e,

he
se ies

,~

a
n

.

x
n

is
Mac-
n >
1

-
n
> 1
key-con e gen
.
Then
(xn)n
>
1is
bounded
.
P oo
.
(i)
.
Take
a
bounded
disk
B
con aining
(x
n
)
n
>
l,
and
deno e
is
a
Cauchy sequence
wi h
espec
o 11 . II
B
,
and,
(E,
S
)
being
Mackey-comple e,
i is
Mackey-con e gen
in
(E,
B
) .
(ü)
I
(x
n
)
n
>_
1

is
unbounded_ he e
is a
bo ni o ous
subse

V
which
does
no
abso b
his
sequence
.
Replacing
(xn
)
n

by a
sub-
á
1
sequencei
necessa y
n
i s
gauge
.
The
sequenceo
pa ial
sums

n a
k
x
k
)
,
'k-1

n
>
1
we
can
suppose
ha
x
11 EZ
n
2
V
o
e e y
n
.
Then
(n
~2
.
x
n
)

is
no
Mackey-con e gen
o
ze o
.
//
na
1
As we ha e
p e iously
announced,be will
show,
h ough
a
coun-
e example,
ha
he
assump ionon
(E,
S
)
in
pa
(ii)
o
he
p e-
ceding
P oposi ion
is
no
supe luous
.
Example
4
.
Le
E
be
he
space
o
measu ab'le
eal
unc ions
on
he
uni
in e al
I =
-
10,1
(wi h
he
s anda d
iden i ica ion),
p o ided
wi h
he
o de
bo nology
:
a
subse
A
C
E
is
bounded
i
he e
is
some
gE
:E,
wi h
g
a
0
and
I l
<
g
o
e e y
E
A
.
This
bo nology
is
con ex,
bu no
he
associa ed
in a opological
bo nology
.
So
E is
no
in a opological
.
De ail
:s
on
his
ac
can
be
' ound
in [3J
.

Mo eo e
he
Mackey-con e gence
ela i e
o
he
o de
bo nology
coincides
wi h
he
almos
e e ywhe e
con e gence
.
We
conside
he
sequence
(
n
)

de ined
as
ollows
:
(we
n
? 1
deno eby
X
T
he
cha ac e is ic
unc iono
a
subse
T o
I)
:
4
= 3
.
X

5
= 3
.
X

}

1

,
and
so on
.
10,1/41

'

(
4
.
z-
ho
Ob iously,
(
)
n

is
bouhded
in he
o de
bo nology
.
Ne e heless,
n ?
1
1

an
n
con e ges
almos e e ywhe ewhen

2
:

lan
1

con e ges
.
l
su ices o
conside
he
case
in
which
a
n
1
0
o all
n
.
Mo eo e ,
i
su ices,
in' his
case,
o
p o e
ha

a
n n
con e ges
in
n
'=
1
measu e,
because
o
an inc easing
sequence,
bo h
ypes
o
con e -
gence
a e
equi alen
.
Fo
each
n, we
deno e
:
Take
a
ixed
n
.
On each in e al
b
n
=

aj

gn =

aj j
2n-1
é
j
<2
n

2n-1
i
j<
2n
he
cons an alue
na
j
o
some
j,
2
n-1
5 j
<
2n
.
The e o e,
gn
exceeds
bn
on his
in e al
i
and
only i
a
j
>

n
1 b
n
.
Bu
his
happens
o
a
mos
n-1
o
hese
in e als
.
Thus
:
2
n-1
Take
now
an
a bi a y
E
>
0,
and
choose
N
such ha
:
b
n
<

n-1

<
E
.
2
n
2
N

n2N
k

k+11

he
unc ion
g
akes
2n
'
2n

n

Then
:
m(
{
x
:

1

9n
(X)
>
E })

<

m(
{

x
:g
n
(x)

>

b
n
})_`Z

n

<

E
.
n
Z
N

n > N

Zn-1
n
=N
This a gumen p o es ha

gn
con e ges
in
measu e,
and
so
n?
1
does

aj
j
.//
j
21
R
EFERENC
E
S
1
.
M
.
A
.
Canela
.
Bo nologí
a
equicon inua
en un
espacio
de
aplicaciones
lineales
con inuas
.
Re is a
de
la
Uni e sidad
de San ande ,
núm
.2,
pa e
1
.
(1979) (111-121)
.
2
.
M
.
A
.
Canela
.
Linea
bo nologies
and
associa ed
opologies
.
Collec
.
Ma h
.
32
(1981),
165-178
.
3
.
M
.
A
.
Canela
.
Som
e
classes
o linea
bo nologlcal
spaces
.
oc
.
Royal
Soc
.
Edinb
.
90
(1981),
155-161
.
4
.
í
.
Hogbe-Nlend
.
Théo ie
des
bo nologies
e
applica ions
.
Lec u e
No es
in
Ma hema ics
213
(Be lin
:
Sp inge ,
1971)
.
5
.
H
.
Hogbe-Nlend
.
Bo nology
and
unc ional
analysis
(Ams e dam
:
No h
Holland,
1978)
.
6
.
V
.
B
.
Mosca elli
.
Bases in
bo nologícal
spaces
.
S udiaMa h
.
50
(1974),
251-264
.
Rebu
el
día
27 de
.
juny
de 1986
Uni e si a
de
Ba celona
Dep
.
d
e
TeO ia
de
Funcions
Facul a de
Ma ema iques
G an
Via,
585
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