<s>
In	O
functional	B-Application
analysis	I-Application
,	O
two	O
methods	O
of	O
constructing	O
normed	O
spaces	O
from	O
disks	O
were	O
systematically	O
employed	O
by	O
Alexander	O
Grothendieck	O
to	O
define	O
nuclear	B-Algorithm
operators	I-Algorithm
and	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
If	O
the	O
disk	O
is	O
both	O
bounded	B-Algorithm
and	O
absorbing	B-Algorithm
then	O
the	O
two	O
auxiliary	B-Algorithm
normed	I-Algorithm
spaces	I-Algorithm
are	O
canonically	O
isomorphic	O
(	O
as	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
and	O
as	O
normed	O
spaces	O
)	O
.	O
</s>
<s>
This	O
justifies	O
the	O
study	O
of	O
Banach	B-Algorithm
disks	I-Algorithm
and	O
is	O
part	O
of	O
the	O
reason	O
why	O
they	O
play	O
an	O
important	O
role	O
in	O
the	O
theory	O
of	O
nuclear	B-Algorithm
operators	I-Algorithm
and	O
nuclear	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
induced	O
by	O
this	O
seminorm	O
is	O
the	O
topology	O
that	O
was	O
defined	O
before	O
.	O
</s>
<s>
If	O
its	O
shown	O
that	O
is	O
a	O
Banach	O
space	O
then	O
will	O
be	O
a	O
Banach	B-Algorithm
disk	I-Algorithm
in	O
TVS	O
that	O
contains	O
as	O
a	O
bounded	B-Algorithm
subset	O
.	O
</s>
<s>
The	O
following	O
result	O
explains	O
why	O
Banach	B-Algorithm
disks	I-Algorithm
are	O
required	O
to	O
be	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
there	O
exists	O
a	O
Hausdorff	O
TVS	O
topology	O
on	O
such	O
that	O
is	O
bounded	B-Algorithm
in	O
then	O
is	O
a	O
norm	O
.	O
</s>
<s>
If	O
is	O
a	O
disk	O
in	O
a	O
vector	O
space	O
and	O
if	O
there	O
exists	O
a	O
TVS	O
topology	O
on	O
such	O
that	O
is	O
a	O
closed	O
and	O
bounded	B-Algorithm
subset	O
of	O
then	O
is	O
the	O
closed	O
unit	O
ball	O
of	O
(	O
that	O
is	O
,	O
)	O
(	O
see	O
footnote	O
for	O
proof	O
)	O
.	O
</s>
<s>
Once	O
this	O
is	O
established	O
,	O
will	O
be	O
a	O
Banach	B-Algorithm
disk	I-Algorithm
in	O
any	O
TVS	O
in	O
which	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
A	O
sequentially	B-Algorithm
complete	I-Algorithm
bounded	B-Algorithm
disk	O
in	O
a	O
Hausdorff	O
TVS	O
is	O
a	O
Banach	B-Algorithm
disk	I-Algorithm
.	O
</s>
<s>
Any	O
disk	O
in	O
a	O
Hausdorff	O
TVS	O
that	O
is	O
complete	O
and	O
bounded	B-Algorithm
(	O
e.g.	O
</s>
<s>
compact	O
)	O
is	O
a	O
Banach	B-Algorithm
disk	I-Algorithm
.	O
</s>
<s>
The	O
closed	O
unit	O
ball	O
in	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
sequentially	B-Algorithm
complete	I-Algorithm
and	O
thus	O
a	O
Banach	B-Algorithm
disk	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
convex	O
balanced	O
closed	O
neighborhood	O
of	O
the	O
origin	O
in	O
then	O
the	O
collection	O
of	O
all	O
neighborhoods	O
where	O
ranges	O
over	O
the	O
positive	O
real	O
numbers	O
,	O
induces	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
topology	O
on	O
When	O
has	O
this	O
topology	O
,	O
it	O
is	O
denoted	O
by	O
Since	O
this	O
topology	O
is	O
not	O
necessarily	O
Hausdorff	O
nor	O
complete	O
,	O
the	O
completion	O
of	O
the	O
Hausdorff	O
space	O
is	O
denoted	O
by	O
so	O
that	O
is	O
a	O
complete	O
Hausdorff	O
space	O
and	O
is	O
a	O
norm	O
on	O
this	O
space	O
making	O
into	O
a	O
Banach	O
space	O
.	O
</s>
<s>
The	O
polar	B-Algorithm
of	O
is	O
a	O
weakly	O
compact	O
bounded	B-Algorithm
equicontinuous	O
disk	O
in	O
and	O
so	O
is	O
infracomplete	B-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
and	O
is	O
a	O
convex	O
balanced	O
and	O
radial	O
set	O
.	O
</s>
<s>
If	O
in	O
addition	O
is	O
bounded	B-Algorithm
in	O
then	O
the	O
seminorm	O
is	O
a	O
norm	O
so	O
in	O
particular	O
,	O
</s>
<s>
In	O
this	O
case	O
,	O
we	O
take	O
to	O
be	O
the	O
vector	O
space	O
instead	O
of	O
so	O
that	O
the	O
notation	O
is	O
unambiguous	O
(	O
whether	O
denotes	O
the	O
space	O
induced	O
by	O
a	O
radial	O
disk	O
or	O
the	O
space	O
induced	O
by	O
a	O
bounded	B-Algorithm
disk	O
)	O
.	O
</s>
<s>
Suppose	O
that	O
in	O
addition	O
and	O
are	O
bounded	B-Algorithm
disks	O
in	O
with	O
so	O
that	O
and	O
the	O
inclusion	O
is	O
a	O
continuous	O
linear	O
map	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
bounded	B-Algorithm
radial	O
disk	O
.	O
</s>
<s>
Since	O
is	O
a	O
bounded	B-Algorithm
disk	O
,	O
if	O
then	O
we	O
may	O
create	O
the	O
auxiliary	B-Algorithm
normed	I-Algorithm
space	I-Algorithm
with	O
norm	O
;	O
since	O
is	O
radial	O
,	O
</s>
<s>
Thus	O
,	O
in	O
this	O
case	O
the	O
two	O
auxiliary	B-Algorithm
normed	I-Algorithm
spaces	I-Algorithm
produced	O
by	O
these	O
two	O
different	O
methods	O
result	O
in	O
the	O
same	O
normed	O
space	O
.	O
</s>
<s>
A	O
disk	O
in	O
a	O
TVS	O
is	O
called	O
infrabornivorous	B-Algorithm
if	O
it	O
absorbs	B-Algorithm
all	O
Banach	B-Algorithm
disks	I-Algorithm
.	O
</s>
<s>
A	O
linear	O
map	O
between	O
two	O
TVSs	O
is	O
called	O
infrabounded	O
if	O
it	O
maps	O
Banach	B-Algorithm
disks	I-Algorithm
to	O
bounded	B-Algorithm
disks	O
.	O
</s>
