<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
in	O
a	O
vector	O
space	O
is	O
a	O
set	O
which	O
can	O
be	O
"	O
inflated	O
"	O
or	O
"	O
scaled	O
up	O
"	O
to	O
eventually	O
always	O
include	O
any	O
given	O
point	O
of	O
the	O
vector	O
space	O
.	O
</s>
<s>
Alternative	O
terms	O
are	O
radial	O
or	O
absorbent	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Every	O
neighborhood	O
of	O
the	O
origin	O
in	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
an	O
absorbing	O
subset	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
definition	O
of	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
(	O
given	O
below	O
)	O
is	O
also	O
tied	O
to	O
this	O
topology	O
.	O
</s>
<s>
If	O
(	O
a	O
necessary	O
condition	O
for	O
to	O
be	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
,	O
or	O
to	O
be	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
a	O
topology	O
)	O
then	O
this	O
list	O
can	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
A	O
subset	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
if	O
it	O
is	O
absorbed	O
by	O
every	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
set	O
is	O
called	O
if	O
it	O
absorbs	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
there	O
does	O
exist	O
non-empty	O
subset	O
of	O
that	O
is	O
absorbed	O
by	O
the	O
unit	O
circle	O
In	O
contrast	O
,	O
every	O
neighborhood	O
of	O
the	O
origin	O
absorbs	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
(	O
and	O
so	O
in	O
particular	O
,	O
absorbs	O
every	O
singleton	O
subset/point	O
)	O
.	O
</s>
<s>
So	O
in	O
particular	O
,	O
can	O
not	O
be	O
absorbing	O
if	O
Every	O
absorbing	B-Algorithm
set	I-Algorithm
must	O
contain	O
the	O
origin	O
.	O
</s>
<s>
contains	O
the	O
origin	O
and	O
for	O
every	O
1-dimensional	O
vector	O
subspace	O
of	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
when	O
is	O
given	O
its	O
unique	O
Hausdorff	O
vector	B-Architecture
topology	I-Architecture
.	O
</s>
<s>
Connection	O
to	O
vector/TVS	O
topologies	O
:	O
This	O
condition	O
gives	O
insight	O
as	O
to	O
why	O
every	O
neighborhood	O
of	O
the	O
origin	O
in	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
necessarily	O
absorbing	O
:	O
If	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
a	O
TVS	O
then	O
for	O
every	O
1-dimensional	O
vector	O
subspace	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
when	O
is	O
endowed	O
with	O
the	O
subspace	O
topology	O
induced	O
on	O
it	O
by	O
This	O
subspace	O
topology	O
is	O
always	O
a	O
vector	O
topologyA	O
topology	O
on	O
a	O
vector	O
space	O
is	O
called	O
a	O
or	O
a	O
if	O
its	O
makes	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
continuous	O
when	O
the	O
scalar	O
field	O
is	O
given	O
its	O
usual	O
norm-induced	O
Euclidean	O
topology	O
(	O
that	O
norm	O
being	O
the	O
absolute	O
value	O
)	O
.	O
</s>
<s>
Thus	O
the	O
subspace	O
topology	O
that	O
any	O
vector	O
subspace	O
inherits	O
from	O
a	O
TVS	O
will	O
once	O
again	O
be	O
a	O
vector	B-Architecture
topology	I-Architecture
.	O
</s>
<s>
So	O
regardless	O
of	O
which	O
of	O
these	O
vector	O
topologies	O
is	O
on	O
the	O
set	O
will	O
be	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
with	O
respect	O
to	O
its	O
unique	O
Hausdorff	O
vector	B-Architecture
topology	I-Architecture
(	O
the	O
Euclidean	O
topology	O
)	O
.If	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
a	O
TVS	O
then	O
it	O
would	O
be	O
pathological	O
if	O
there	O
existed	O
any	O
1-dimensional	O
vector	O
subspace	O
in	O
which	O
was	O
not	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
at	O
least	O
TVS	O
topology	O
on	O
The	O
only	O
TVS	O
topologies	O
on	O
are	O
the	O
Hausdorff	O
Euclidean	O
topology	O
and	O
the	O
trivial	O
topology	O
,	O
which	O
is	O
a	O
subset	O
of	O
the	O
Euclidean	O
topology	O
.	O
</s>
<s>
For	O
this	O
to	O
happen	O
,	O
it	O
suffices	O
for	O
to	O
be	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
that	O
is	O
also	O
convex	O
balanced	O
and	O
closed	O
in	O
(	O
such	O
a	O
set	O
is	O
called	O
a	O
and	O
it	O
will	O
be	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
because	O
like	O
every	O
finite-dimensional	O
Euclidean	O
space	O
,	O
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
Give	O
its	O
unique	O
Hausdorff	O
vector	B-Architecture
topology	I-Architecture
so	O
it	O
remains	O
to	O
show	O
that	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
If	O
then	O
we	O
are	O
done	O
,	O
so	O
assume	O
that	O
The	O
set	O
is	O
a	O
union	O
of	O
two	O
intervals	O
,	O
each	O
of	O
which	O
contains	O
an	O
open	O
sub-interval	O
that	O
contains	O
the	O
origin	O
;	O
moreover	O
,	O
the	O
intersection	O
of	O
these	O
two	O
intervals	O
is	O
precisely	O
the	O
origin	O
.	O
</s>
<s>
Any	O
superset	O
of	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
is	O
absorbing	O
.	O
</s>
<s>
Thus	O
the	O
union	O
of	O
any	O
family	O
of	O
(	O
one	O
or	O
more	O
)	O
absorbing	B-Algorithm
sets	I-Algorithm
is	O
absorbing	O
.	O
</s>
<s>
The	O
image	O
of	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
under	O
a	O
surjective	O
linear	O
operator	O
is	O
again	O
absorbing	O
.	O
</s>
<s>
The	O
intersection	O
of	O
a	O
finite	O
family	O
of	O
(	O
one	O
or	O
more	O
)	O
absorbing	B-Algorithm
sets	I-Algorithm
is	O
absorbing	O
.	O
</s>
<s>
This	O
guarantees	O
that	O
the	O
Minkowski	O
functional	O
of	O
will	O
be	O
a	O
seminorm	O
on	O
thereby	O
making	O
into	O
a	O
seminormed	O
space	O
that	O
carries	O
its	O
canonical	O
pseduometrizable	B-Algorithm
topology	O
.	O
</s>
<s>
The	O
set	O
of	O
scalar	O
multiples	O
as	O
ranges	O
over	O
(	O
or	O
over	O
any	O
other	O
set	O
of	O
non-zero	O
scalars	O
having	O
as	O
a	O
limit	O
point	O
)	O
forms	O
a	O
neighborhood	O
basis	O
of	O
absorbing	O
disks	O
at	O
the	O
origin	O
for	O
this	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
and	O
if	O
this	O
convex	O
absorbing	O
subset	O
is	O
also	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
then	O
the	O
same	O
will	O
be	O
true	O
of	O
the	O
absorbing	O
disk	O
in	O
which	O
case	O
will	O
be	O
a	O
norm	O
and	O
will	O
form	O
what	O
is	O
known	O
as	O
an	O
auxiliary	B-Algorithm
normed	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
absorbing	B-Algorithm
set	I-Algorithm
contains	O
the	O
origin	O
.	O
</s>
