<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
set	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
bounded	B-Algorithm
or	O
von	B-Algorithm
Neumann	I-Algorithm
bounded	I-Algorithm
,	O
if	O
every	O
neighborhood	O
of	O
the	O
zero	O
vector	O
can	O
be	O
inflated	O
to	O
include	O
the	O
set	O
.	O
</s>
<s>
A	O
set	O
that	O
is	O
not	O
bounded	B-Algorithm
is	O
called	O
unbounded	O
.	O
</s>
<s>
Bounded	B-Algorithm
sets	I-Algorithm
are	O
a	O
natural	O
way	O
to	O
define	O
locally	B-Algorithm
convex	I-Algorithm
polar	B-Algorithm
topologies	I-Algorithm
on	O
the	O
vector	O
spaces	O
in	O
a	O
dual	B-Algorithm
pair	I-Algorithm
,	O
as	O
the	O
polar	B-Algorithm
set	I-Algorithm
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
is	O
an	O
absolutely	O
convex	O
and	O
absorbing	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
is	O
absorbed	B-Algorithm
by	O
every	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Statement	O
(	O
2	O
)	O
may	O
become	O
:	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
is	O
absorbed	B-Algorithm
by	O
every	O
balanced	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
If	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
the	O
adjective	O
"	O
convex	O
"	O
may	O
be	O
also	O
be	O
added	O
to	O
any	O
of	O
these	O
5	O
replacements	O
.	O
</s>
<s>
This	O
was	O
the	O
definition	O
of	O
"	O
bounded	B-Algorithm
"	O
that	O
Andrey	O
Kolmogorov	O
used	O
in	O
1934	O
,	O
which	O
is	O
the	O
same	O
as	O
the	O
definition	O
introduced	O
by	O
Stanisław	O
Mazur	O
and	O
Władysław	O
Orlicz	O
in	O
1933	O
for	O
metrizable	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Kolmogorov	O
used	O
this	O
definition	O
to	O
prove	O
that	O
a	O
TVS	O
is	O
seminormable	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
bounded	B-Algorithm
convex	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Every	O
countable	O
subset	O
of	O
is	O
bounded	B-Algorithm
(	O
according	O
to	O
any	O
defining	O
condition	O
other	O
than	O
this	O
one	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
whose	O
topology	O
is	O
defined	O
by	O
a	O
family	O
of	O
continuous	O
seminorms	O
,	O
then	O
this	O
list	O
may	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
There	O
exists	O
a	O
sequence	O
of	O
non-zero	O
scalars	O
such	O
that	O
for	O
every	O
sequence	O
in	O
the	O
sequence	O
is	O
bounded	B-Algorithm
in	O
(	O
according	O
to	O
any	O
defining	O
condition	O
other	O
than	O
this	O
one	O
)	O
.	O
</s>
<s>
A	O
subset	O
that	O
is	O
not	O
bounded	B-Algorithm
is	O
called	O
.	O
</s>
<s>
In	O
any	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
,	O
the	O
set	O
of	O
closed	O
and	O
bounded	B-Algorithm
disks	O
are	O
a	O
base	O
of	O
bounded	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Unless	O
indicated	O
otherwise	O
,	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
need	O
not	O
be	O
Hausdorff	O
nor	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Finite	O
sets	O
are	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
totally	O
bounded	B-Algorithm
subset	O
of	O
a	O
TVS	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
relatively	O
compact	O
set	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
points	O
of	O
a	O
Cauchy	O
sequence	O
is	O
bounded	B-Algorithm
,	O
the	O
set	O
of	O
points	O
of	O
a	O
Cauchy	O
net	O
need	O
not	O
be	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
closure	O
of	O
the	O
origin	O
(	O
referring	O
to	O
the	O
closure	O
of	O
the	O
set	O
)	O
is	O
always	O
a	O
bounded	B-Algorithm
closed	O
vector	O
subspace	O
.	O
</s>
<s>
A	O
set	O
that	O
is	O
not	O
bounded	B-Algorithm
is	O
said	O
to	O
be	O
unbounded	O
.	O
</s>
<s>
In	O
any	O
TVS	O
,	O
finite	O
unions	O
,	O
finite	O
Minkowski	B-Algorithm
sums	I-Algorithm
,	O
scalar	O
multiples	O
,	O
translations	O
,	O
subsets	O
,	O
closures	O
,	O
interiors	O
,	O
and	O
balanced	O
hulls	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
are	O
again	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
any	O
locally	B-Algorithm
convex	I-Algorithm
TVS	I-Algorithm
,	O
the	O
convex	O
hull	O
(	O
also	O
called	O
the	O
convex	O
envelope	O
)	O
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
is	O
again	O
bounded	B-Algorithm
.	O
</s>
<s>
However	O
,	O
this	O
may	O
be	O
false	O
if	O
the	O
space	O
is	O
not	O
locally	B-Algorithm
convex	I-Algorithm
,	O
as	O
the	O
(	O
non-locally	O
convex	O
)	O
Lp	O
space	O
spaces	O
for	O
have	O
no	O
nontrivial	O
open	O
convex	O
subsets	O
.	O
</s>
<s>
The	O
image	O
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
under	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
is	O
a	O
bounded	B-Algorithm
subset	O
of	O
the	O
codomain	O
.	O
</s>
<s>
A	O
subset	O
of	O
an	O
arbitrary	O
(	O
Cartesian	O
)	O
product	O
of	O
TVSs	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
its	O
image	O
under	O
every	O
coordinate	O
projections	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
has	O
a	O
bounded	B-Algorithm
neighborhood	O
of	O
zero	O
if	O
and	O
only	O
if	O
its	O
topology	O
can	O
be	O
defined	O
by	O
a	O
seminorm	O
.	O
</s>
<s>
The	O
polar	B-Algorithm
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
is	O
an	O
absolutely	O
convex	O
and	O
absorbing	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
definition	O
of	O
uniformly	O
bounded	B-Algorithm
sets	I-Algorithm
given	O
below	O
,	O
Mackey	O
's	O
countability	O
condition	O
can	O
be	O
restated	O
as	O
:	O
If	O
are	O
bounded	B-Algorithm
subsets	O
of	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
there	O
exists	O
a	O
sequence	O
of	O
positive	O
real	O
numbers	O
such	O
that	O
are	O
uniformly	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
words	O
,	O
given	O
any	O
countable	O
family	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
in	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
it	O
is	O
possible	O
to	O
scale	O
each	O
set	O
by	O
its	O
own	O
positive	O
real	O
so	O
that	O
they	O
become	O
uniformly	O
bounded	B-Algorithm
.	O
</s>
<s>
So	O
for	O
every	O
and	O
every	O
which	O
implies	O
that	O
Thus	O
is	O
bounded	B-Algorithm
in	O
Q.E.D.	O
</s>
<s>
If	O
then	O
being	O
bounded	B-Algorithm
guarantees	O
the	O
existence	O
of	O
some	O
positive	O
integer	O
such	O
that	O
where	O
the	O
linearity	O
of	O
every	O
now	O
implies	O
thus	O
and	O
hence	O
as	O
desired	O
.	O
</s>
<s>
It	O
will	O
be	O
shown	O
that	O
for	O
every	O
thus	O
demonstrating	O
that	O
is	O
uniformly	O
bounded	B-Algorithm
in	O
and	O
completing	O
the	O
proof	O
.	O
</s>
<s>
The	O
definition	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
can	O
be	O
generalized	O
to	O
topological	O
modules	O
.	O
</s>
