<s>
In	O
mathematics	O
,	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
also	O
called	O
a	O
linear	B-Architecture
topological	I-Architecture
space	I-Architecture
and	O
commonly	O
abbreviated	O
TVS	O
or	O
t.v.s.	O
)	O
</s>
<s>
is	O
one	O
of	O
the	O
basic	O
structures	O
investigated	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
vector	O
space	O
that	O
is	O
also	O
a	O
topological	O
space	O
with	O
the	O
property	O
that	O
the	O
vector	O
space	O
operations	O
(	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
)	O
are	O
also	O
continuous	O
functions	O
.	O
</s>
<s>
Such	O
a	O
topology	O
is	O
called	O
a	O
and	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
a	O
uniform	O
topological	O
structure	O
,	O
allowing	O
a	O
notion	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
and	O
completeness	B-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
most	O
widely	O
studied	O
categories	O
of	O
TVSs	O
are	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
This	O
article	O
focuses	O
on	O
TVSs	O
that	O
are	O
not	O
necessarily	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Many	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
are	O
spaces	O
of	O
functions	O
,	O
or	O
linear	B-Architecture
operators	I-Architecture
acting	O
on	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
and	O
the	O
topology	O
is	O
often	O
defined	O
so	O
as	O
to	O
capture	O
a	O
particular	O
notion	O
of	O
convergence	O
of	O
sequences	O
of	O
functions	O
.	O
</s>
<s>
In	O
this	O
article	O
,	O
the	O
scalar	O
field	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
will	O
be	O
assumed	O
to	O
be	O
either	O
the	O
complex	O
numbers	O
or	O
the	O
real	O
numbers	O
unless	O
clearly	O
stated	O
otherwise	O
.	O
</s>
<s>
This	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
because	O
:	O
</s>
<s>
Thus	O
all	O
Banach	O
spaces	O
and	O
Hilbert	O
spaces	O
are	O
examples	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
There	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
whose	O
topology	O
is	O
not	O
induced	O
by	O
a	O
norm	O
,	O
but	O
are	O
still	O
of	O
interest	O
in	O
analysis	O
.	O
</s>
<s>
Examples	O
of	O
such	O
spaces	O
are	O
spaces	O
of	O
holomorphic	O
functions	O
on	O
an	O
open	O
domain	O
,	O
spaces	O
of	O
infinitely	O
differentiable	O
functions	O
,	O
the	O
Schwartz	B-Algorithm
spaces	I-Algorithm
,	O
and	O
spaces	O
of	O
test	O
functions	O
and	O
the	O
spaces	O
of	O
distributions	O
on	O
them	O
.	O
</s>
<s>
These	O
are	O
all	O
examples	O
of	O
Montel	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
An	O
infinite-dimensional	O
Montel	B-Algorithm
space	I-Algorithm
is	O
never	O
normable	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
norm	O
for	O
a	O
given	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
characterized	O
by	O
Kolmogorov	O
's	O
normability	O
criterion	O
.	O
</s>
<s>
A	O
topological	O
field	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
each	O
of	O
its	O
subfields	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
a	O
vector	O
space	O
over	O
a	O
topological	O
field	O
(	O
most	O
often	O
the	O
real	O
or	O
complex	O
numbers	O
with	O
their	O
standard	O
topologies	O
)	O
that	O
is	O
endowed	O
with	O
a	O
topology	O
such	O
that	O
vector	O
addition	O
and	O
scalar	O
multiplication	O
are	O
continuous	O
functions	O
(	O
where	O
the	O
domains	O
of	O
these	O
functions	O
are	O
endowed	O
with	O
product	O
topologies	O
)	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
also	O
a	O
commutative	O
topological	O
group	O
under	O
addition	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
said	O
to	O
be	O
if	O
it	O
is	O
Hausdorff	O
;	O
importantly	O
,	O
"	O
separated	O
"	O
does	O
not	O
mean	O
separable	O
.	O
</s>
<s>
The	O
topological	O
and	O
linear	B-Architecture
algebraic	O
structures	O
can	O
be	O
tied	O
together	O
even	O
more	O
closely	O
with	O
additional	O
assumptions	O
,	O
the	O
most	O
common	O
of	O
which	O
are	O
listed	O
below	O
.	O
</s>
<s>
The	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
over	O
a	O
given	O
topological	O
field	O
is	O
commonly	O
denoted	O
or	O
The	O
objects	O
are	O
the	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
over	O
and	O
the	O
morphisms	O
are	O
the	O
continuous	O
-linear	O
maps	O
from	O
one	O
object	O
to	O
another	O
.	O
</s>
<s>
A	O
(	O
abbreviated	O
)	O
,	O
also	O
called	O
a	O
,	O
is	O
an	O
injective	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
Equivalently	O
,	O
a	O
TVS-embedding	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
that	O
is	O
also	O
a	O
topological	O
embedding	O
.	O
</s>
<s>
A	O
(	O
abbreviated	O
)	O
,	O
also	O
called	O
a	O
or	O
an	O
,	O
is	O
a	O
bijective	O
linear	B-Architecture
homeomorphism	O
.	O
</s>
<s>
Many	O
properties	O
of	O
TVSs	O
that	O
are	O
studied	O
,	O
such	O
as	O
local	B-Algorithm
convexity	I-Algorithm
,	O
metrizability	B-Algorithm
,	O
completeness	B-Algorithm
,	O
and	O
normability	O
,	O
are	O
invariant	O
under	O
TVS	O
isomorphisms	O
.	O
</s>
<s>
Since	O
every	O
vector	O
topology	O
is	O
translation	B-Algorithm
invariant	O
(	O
which	O
means	O
that	O
for	O
all	O
the	O
map	O
defined	O
by	O
is	O
a	O
homeomorphism	O
)	O
,	O
to	O
define	O
a	O
vector	O
topology	O
it	O
suffices	O
to	O
define	O
a	O
neighborhood	O
basis	O
(	O
or	O
subbasis	O
)	O
for	O
it	O
at	O
the	O
origin	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
set	O
of	O
all	O
balanced	O
and	O
absorbing	B-Algorithm
subsets	O
of	O
a	O
vector	O
space	O
does	O
not	O
satisfy	O
the	O
conditions	O
of	O
this	O
theorem	O
and	O
does	O
not	O
form	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
for	O
any	O
vector	O
topology	O
.	O
</s>
<s>
absorbing	B-Algorithm
,	O
closed	O
,	O
convex	O
,	O
open	O
,	O
symmetric	O
,	O
barrelled	B-Algorithm
,	O
absolutely	O
convex/disked	O
,	O
etc	O
.	O
)	O
</s>
<s>
if	O
is	O
summative	O
,	O
absorbing	B-Algorithm
,	O
and	O
balanced	O
.	O
</s>
<s>
If	O
is	O
an	O
absorbing	B-Algorithm
disk	O
in	O
a	O
vector	O
space	O
then	O
the	O
sequence	O
defined	O
by	O
forms	O
a	O
string	O
beginning	O
with	O
This	O
is	O
called	O
the	O
natural	O
string	O
of	O
Moreover	O
,	O
if	O
a	O
vector	O
space	O
has	O
countable	O
dimension	O
then	O
every	O
string	O
contains	O
an	O
absolutely	O
convex	O
string	O
.	O
</s>
<s>
These	O
functions	O
can	O
then	O
be	O
used	O
to	O
prove	O
many	O
of	O
the	O
basic	O
properties	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
A	O
proof	O
of	O
the	O
above	O
theorem	O
is	O
given	O
in	O
the	O
article	O
on	O
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Defining	O
vector	O
topologies	O
using	O
collections	O
of	O
strings	O
is	O
particularly	O
useful	O
for	O
defining	O
classes	O
of	O
TVSs	O
that	O
are	O
not	O
necessarily	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
If	O
is	O
the	O
set	O
of	O
all	O
topological	O
strings	O
in	O
a	O
TVS	O
then	O
A	O
Hausdorff	O
TVS	O
is	O
metrizable	B-Algorithm
if	O
and	O
only	O
if	O
its	O
topology	O
can	O
be	O
induced	O
by	O
a	O
single	O
topological	O
string	O
.	O
</s>
<s>
A	O
vector	O
space	O
is	O
an	O
abelian	O
group	O
with	O
respect	O
to	O
the	O
operation	O
of	O
addition	O
,	O
and	O
in	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
the	O
inverse	O
operation	O
is	O
always	O
continuous	O
(	O
since	O
it	O
is	O
the	O
same	O
as	O
multiplication	O
by	O
)	O
.	O
</s>
<s>
Hence	O
,	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
an	O
abelian	O
topological	O
group	O
.	O
</s>
<s>
Let	O
be	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
Given	O
a	O
subspace	O
the	O
quotient	O
space	O
with	O
the	O
usual	O
quotient	O
topology	O
is	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
if	O
and	O
only	O
if	O
is	O
closed	O
.	O
</s>
<s>
for	O
all	O
the	O
map	O
defined	O
by	O
is	O
a	O
homeomorphism	O
,	O
but	O
if	O
then	O
it	O
is	O
not	O
linear	B-Architecture
and	O
so	O
not	O
a	O
TVS-isomorphism	O
.	O
</s>
<s>
This	O
means	O
that	O
if	O
then	O
the	O
linear	B-Architecture
map	I-Architecture
defined	O
by	O
is	O
a	O
homeomorphism	O
.	O
</s>
<s>
Using	O
produces	O
the	O
negation	O
map	O
defined	O
by	O
which	O
is	O
consequently	O
a	O
linear	B-Architecture
homeomorphism	O
and	O
thus	O
a	O
TVS-isomorphism	O
.	O
</s>
<s>
Every	O
neighborhood	O
of	O
the	O
origin	O
is	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
and	O
contains	O
an	O
open	O
balanced	O
neighborhood	O
of	O
so	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
a	O
local	O
base	O
of	O
absorbing	B-Algorithm
and	O
balanced	O
sets	O
.	O
</s>
<s>
The	O
origin	O
even	O
has	O
a	O
neighborhood	O
basis	O
consisting	O
of	O
closed	O
balanced	O
neighborhoods	O
of	O
if	O
the	O
space	O
is	O
locally	B-Algorithm
convex	I-Algorithm
then	O
it	O
also	O
has	O
a	O
neighborhood	O
basis	O
consisting	O
of	O
closed	O
convex	O
balanced	O
neighborhoods	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
subset	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
bounded	B-Algorithm
if	O
for	O
every	O
neighborhood	O
of	O
the	O
origin	O
,	O
then	O
when	O
is	O
sufficiently	O
large	O
.	O
</s>
<s>
The	O
definition	O
of	O
boundedness	B-Algorithm
can	O
be	O
weakened	O
a	O
bit	O
;	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
every	O
countable	O
subset	O
of	O
it	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
A	O
set	O
is	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
each	O
of	O
its	O
subsequences	O
is	O
a	O
bounded	B-Algorithm
set	I-Algorithm
.	O
</s>
<s>
Every	O
totally	O
bounded	B-Algorithm
set	I-Algorithm
is	O
bounded	B-Algorithm
.	O
</s>
<s>
A	O
TVS	O
is	O
pseudometrizable	B-Algorithm
if	O
and	O
only	O
if	O
it	O
has	O
a	O
countable	O
neighborhood	O
basis	O
at	O
the	O
origin	O
,	O
or	O
equivalent	O
,	O
if	O
and	O
only	O
if	O
its	O
topology	O
is	O
generated	O
by	O
an	O
F-seminorm	O
.	O
</s>
<s>
A	O
TVS	O
is	O
metrizable	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
Hausdorff	O
and	O
pseudometrizable	B-Algorithm
.	O
</s>
<s>
More	O
strongly	O
:	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
said	O
to	O
be	O
normable	O
if	O
its	O
topology	O
can	O
be	O
induced	O
by	O
a	O
norm	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
normable	O
if	O
and	O
only	O
if	O
it	O
is	O
Hausdorff	O
and	O
has	O
a	O
convex	O
bounded	B-Algorithm
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Every	O
TVS	O
is	O
assumed	O
to	O
be	O
endowed	O
with	O
this	O
canonical	B-Algorithm
uniformity	I-Algorithm
,	O
which	O
makes	O
all	O
TVSs	O
into	O
uniform	O
spaces	O
.	O
</s>
<s>
This	O
allows	O
one	O
to	O
about	O
related	O
notions	O
such	O
as	O
completeness	B-Algorithm
,	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
Cauchy	O
nets	O
,	O
and	O
uniform	O
continuity	O
.	O
</s>
<s>
This	O
implies	O
that	O
every	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
Tychonoff	O
.	O
</s>
<s>
A	O
subspace	O
of	O
a	O
TVS	O
is	O
compact	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
and	O
totally	O
bounded	B-Algorithm
(	O
for	O
Hausdorff	O
TVSs	O
,	O
a	O
set	O
being	O
totally	O
bounded	B-Algorithm
is	O
equivalent	O
to	O
it	O
being	O
precompact	O
)	O
.	O
</s>
<s>
Every	O
Cauchy	O
sequence	O
is	O
bounded	B-Algorithm
,	O
although	O
Cauchy	O
nets	O
and	O
Cauchy	O
filters	O
may	O
not	O
be	O
bounded	B-Algorithm
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
where	O
every	O
Cauchy	O
sequence	O
converges	B-Algorithm
is	O
called	O
sequentially	B-Algorithm
complete	I-Algorithm
;	O
in	O
general	O
,	O
it	O
may	O
not	O
be	O
complete	B-Algorithm
(	O
in	O
the	O
sense	O
that	O
all	O
Cauchy	O
filters	O
converge	O
)	O
.	O
</s>
<s>
Because	O
of	O
this	O
,	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
can	O
be	O
completed	O
and	O
is	O
thus	O
a	O
dense	O
linear	B-Architecture
subspace	O
of	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
TVS	O
has	O
a	O
completion	B-Algorithm
and	O
every	O
Hausdorff	O
TVS	O
has	O
a	O
Hausdorff	O
completion	B-Algorithm
.	O
</s>
<s>
Every	O
TVS	O
(	O
even	O
those	O
that	O
are	O
Hausdorff	O
and/or	O
complete	B-Algorithm
)	O
has	O
infinitely	O
many	O
non-isomorphic	O
non-Hausdorff	O
completions	O
.	O
</s>
<s>
A	O
compact	O
subset	O
of	O
a	O
TVS	O
(	O
not	O
necessarily	O
Hausdorff	O
)	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
A	O
complete	B-Algorithm
subset	O
of	O
a	O
Hausdorff	O
TVS	O
is	O
closed	O
.	O
</s>
<s>
If	O
is	O
a	O
complete	B-Algorithm
subset	O
of	O
a	O
TVS	O
then	O
any	O
subset	O
of	O
that	O
is	O
closed	O
in	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
Any	O
vector	O
space	O
(	O
including	O
those	O
that	O
are	O
infinite	O
dimensional	O
)	O
endowed	O
with	O
the	O
trivial	O
topology	O
is	O
a	O
compact	O
(	O
and	O
thus	O
locally	O
compact	O
)	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
seminormable	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
linear	B-Architecture
map	I-Architecture
from	O
into	O
another	O
TVS	O
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
If	O
has	O
an	O
uncountable	O
Hamel	O
basis	O
then	O
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
metrizable	B-Algorithm
.	O
</s>
<s>
A	O
Cartesian	O
product	O
of	O
a	O
family	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
when	O
endowed	O
with	O
the	O
product	O
topology	O
,	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
With	O
this	O
product	O
topology	O
,	O
becomes	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
whose	O
topology	O
is	O
called	O
The	O
reason	O
for	O
this	O
name	O
is	O
the	O
following	O
:	O
if	O
is	O
a	O
sequence	O
(	O
or	O
more	O
generally	O
,	O
a	O
net	O
)	O
of	O
elements	O
in	O
and	O
if	O
then	O
converges	B-Algorithm
to	O
in	O
if	O
and	O
only	O
if	O
for	O
every	O
real	O
number	O
converges	B-Algorithm
to	O
in	O
This	O
TVS	O
is	O
complete	B-Algorithm
,	O
Hausdorff	O
,	O
and	O
locally	B-Algorithm
convex	I-Algorithm
but	O
not	O
metrizable	B-Algorithm
and	O
consequently	O
not	O
normable	O
;	O
indeed	O
,	O
every	O
neighborhood	O
of	O
the	O
origin	O
in	O
the	O
product	O
topology	O
contains	O
lines	O
(	O
that	O
is	O
,	O
1-dimensional	O
vector	O
subspaces	O
,	O
which	O
are	O
subsets	O
of	O
the	O
form	O
with	O
)	O
.	O
</s>
<s>
By	O
F	B-Algorithm
.	I-Algorithm
Riesz	I-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
finite-dimensional	O
if	O
and	O
only	O
if	O
it	O
is	O
locally	O
compact	O
,	O
which	O
happens	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
compact	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Let	O
be	O
a	O
vector	O
space	O
over	O
of	O
finite	O
dimension	O
and	O
so	O
that	O
is	O
vector	O
space	O
isomorphic	O
to	O
(	O
explicitly	O
,	O
this	O
means	O
that	O
there	O
exists	O
a	O
linear	B-Architecture
isomorphism	I-Architecture
between	O
the	O
vector	O
spaces	O
and	O
)	O
.	O
</s>
<s>
Since	O
the	O
field	O
is	O
itself	O
a	O
-dimensional	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
over	O
and	O
since	O
it	O
plays	O
an	O
important	O
role	O
in	O
the	O
definition	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
this	O
dichotomy	O
plays	O
an	O
important	O
role	O
in	O
the	O
definition	O
of	O
an	O
absorbing	B-Algorithm
set	I-Algorithm
and	O
has	O
consequences	O
that	O
reverberate	O
throughout	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
If	O
is	O
a	O
non-trivial	O
vector	O
space	O
(	O
that	O
is	O
,	O
of	O
non-zero	O
dimension	O
)	O
then	O
the	O
discrete	O
topology	O
on	O
(	O
which	O
is	O
always	O
metrizable	B-Algorithm
)	O
is	O
a	O
TVS	O
topology	O
because	O
despite	O
making	O
addition	O
and	O
negation	O
continuous	O
(	O
which	O
makes	O
it	O
into	O
a	O
topological	O
group	O
under	O
addition	O
)	O
,	O
it	O
fails	O
to	O
make	O
scalar	O
multiplication	O
continuous	O
.	O
</s>
<s>
A	O
linear	B-Architecture
operator	I-Architecture
between	O
two	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
which	O
is	O
continuous	O
at	O
one	O
point	O
is	O
continuous	O
on	O
the	O
whole	O
domain	O
.	O
</s>
<s>
Moreover	O
,	O
a	O
linear	B-Architecture
operator	I-Architecture
is	O
continuous	O
if	O
is	O
bounded	B-Algorithm
(	O
as	O
defined	O
below	O
)	O
for	O
some	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
A	O
hyperplane	O
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
either	O
dense	O
or	O
closed	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
functional	I-Algorithm
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
either	O
dense	O
or	O
closed	O
kernel	O
.	O
</s>
<s>
In	O
fact	O
,	O
several	O
principal	O
results	O
in	O
functional	B-Application
analysis	I-Application
fail	O
to	O
hold	O
in	O
general	O
for	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
:	O
the	O
closed	O
graph	O
theorem	O
,	O
the	O
open	O
mapping	O
theorem	O
,	O
and	O
the	O
fact	O
that	O
the	O
dual	O
space	O
of	O
the	O
space	O
separates	O
points	O
in	O
the	O
space	O
.	O
</s>
<s>
Below	O
are	O
some	O
common	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
,	O
roughly	O
in	O
order	O
of	O
increasing	O
"	O
niceness.	O
"	O
</s>
<s>
F-spaces	B-Algorithm
are	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
with	O
a	O
translation-invariant	O
metric	O
.	O
</s>
<s>
Locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
:	O
here	O
each	O
point	O
has	O
a	O
local	O
base	O
consisting	O
of	O
convex	O
sets	O
.	O
</s>
<s>
By	O
a	O
technique	O
known	O
as	O
Minkowski	O
functionals	O
it	O
can	O
be	O
shown	O
that	O
a	O
space	O
is	O
locally	B-Algorithm
convex	I-Algorithm
if	O
and	O
only	O
if	O
its	O
topology	O
can	O
be	O
defined	O
by	O
a	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
Local	B-Algorithm
convexity	I-Algorithm
is	O
the	O
minimum	O
requirement	O
for	O
"	O
geometrical	O
"	O
arguments	O
like	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
.	O
</s>
<s>
Barrelled	B-Algorithm
spaces	I-Algorithm
:	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
where	O
the	O
Banach	O
–	O
Steinhaus	O
theorem	O
holds	O
.	O
</s>
<s>
Bornological	B-Algorithm
space	I-Algorithm
:	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
where	O
the	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
to	O
any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
are	O
exactly	O
the	O
bounded	B-Algorithm
linear	B-Architecture
operators	I-Architecture
.	O
</s>
<s>
Stereotype	O
space	O
:	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
satisfying	O
a	O
variant	O
of	O
reflexivity	O
condition	O
,	O
where	O
the	O
dual	O
space	O
is	O
endowed	O
with	O
the	O
topology	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
on	O
totally	O
bounded	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
Fréchet	B-Algorithm
spaces	I-Algorithm
:	O
these	O
are	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
where	O
the	O
topology	O
comes	O
from	O
a	O
translation-invariant	O
metric	O
,	O
or	O
equivalently	O
:	O
from	O
a	O
countable	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
Many	O
interesting	O
spaces	O
of	O
functions	O
fall	O
into	O
this	O
class	O
--	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
under	O
the	O
seminorms	O
A	O
locally	B-Algorithm
convex	I-Algorithm
F-space	B-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
LF-spaces	B-Algorithm
are	O
limits	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Nuclear	B-Algorithm
spaces	I-Algorithm
:	O
these	O
are	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
with	O
the	O
property	O
that	O
every	O
bounded	B-Algorithm
map	O
from	O
the	O
nuclear	B-Algorithm
space	I-Algorithm
to	O
an	O
arbitrary	O
Banach	O
space	O
is	O
a	O
nuclear	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Normed	O
spaces	O
and	O
seminormed	O
spaces	O
:	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
where	O
the	O
topology	O
can	O
be	O
described	O
by	O
a	O
single	O
norm	O
or	O
seminorm	O
.	O
</s>
<s>
In	O
normed	O
spaces	O
a	O
linear	B-Architecture
operator	I-Architecture
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
Banach	O
spaces	O
:	O
Complete	B-Algorithm
normed	O
vector	O
spaces	O
.	O
</s>
<s>
Most	O
of	O
functional	B-Application
analysis	I-Application
is	O
formulated	O
for	O
Banach	O
spaces	O
.	O
</s>
<s>
This	O
class	O
includes	O
the	O
spaces	O
with	O
the	O
space	O
of	O
functions	O
of	O
bounded	B-Algorithm
variation	O
,	O
and	O
certain	O
spaces	O
of	O
measures	O
.	O
</s>
<s>
As	O
pointed	O
out	O
in	O
the	O
preceding	O
section	O
,	O
for	O
a	O
given	O
finite	O
there	O
is	O
only	O
one	O
-dimensional	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
up	O
to	O
isomorphism	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
a	O
continuous	O
dual	O
spacethe	O
set	O
of	O
all	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
,	O
that	O
is	O
,	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
from	O
the	O
space	O
into	O
the	O
base	O
field	O
A	O
topology	O
on	O
the	O
dual	O
can	O
be	O
defined	O
to	O
be	O
the	O
coarsest	O
topology	O
such	O
that	O
the	O
dual	O
pairing	O
each	O
point	O
evaluation	O
is	O
continuous	O
.	O
</s>
<s>
This	O
turns	O
the	O
dual	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
very	O
important	O
in	O
applications	O
because	O
of	O
its	O
compactness	O
properties	O
(	O
see	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
)	O
.	O
</s>
<s>
Caution	O
:	O
Whenever	O
is	O
a	O
non-normable	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
pairing	O
map	O
is	O
never	O
continuous	O
,	O
no	O
matter	O
which	O
vector	O
space	O
topology	O
one	O
chooses	O
on	O
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
a	O
non-trivial	O
continuous	O
dual	O
space	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
proper	O
convex	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
The	O
convex	O
hull	O
of	O
a	O
subset	O
is	O
equal	O
to	O
the	O
set	O
of	O
all	O
of	O
elements	O
in	O
which	O
are	O
finite	O
linear	B-Architecture
combinations	O
of	O
the	O
form	O
where	O
is	O
an	O
integer	O
,	O
and	O
sum	O
to	O
The	O
intersection	O
of	O
any	O
family	O
of	O
convex	O
sets	O
is	O
convex	O
and	O
the	O
convex	O
hull	O
of	O
a	O
subset	O
is	O
equal	O
to	O
the	O
intersection	O
of	O
all	O
convex	O
sets	O
that	O
contain	O
it	O
.	O
</s>
<s>
If	O
is	O
an	O
absorbing	B-Algorithm
disk	O
in	O
a	O
TVS	O
and	O
if	O
is	O
the	O
Minkowski	O
functional	O
of	O
then	O
where	O
importantly	O
,	O
it	O
was	O
assumed	O
that	O
had	O
any	O
topological	O
properties	O
nor	O
that	O
was	O
continuous	O
(	O
which	O
happens	O
if	O
and	O
only	O
if	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
TVS	O
that	O
is	O
of	O
the	O
second	O
category	O
in	O
itself	O
(	O
that	O
is	O
,	O
a	O
nonmeager	O
space	O
)	O
then	O
any	O
closed	O
convex	O
absorbing	B-Algorithm
subset	O
of	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Explicitly	O
,	O
this	O
means	O
that	O
if	O
is	O
a	O
convex	O
subset	O
of	O
a	O
TVS	O
(	O
not	O
necessarily	O
Hausdorff	O
or	O
locally	B-Algorithm
convex	I-Algorithm
)	O
,	O
and	O
then	O
the	O
open	O
line	O
segment	O
joining	O
and	O
belongs	O
to	O
the	O
interior	O
of	O
that	O
is	O
,	O
Fix	O
so	O
it	O
remains	O
to	O
show	O
that	O
belongs	O
to	O
By	O
replacing	O
with	O
if	O
necessary	O
,	O
we	O
may	O
assume	O
without	O
loss	O
of	O
generality	O
that	O
and	O
so	O
it	O
remains	O
to	O
show	O
that	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
.	O
</s>
<s>
Let	O
so	O
that	O
Since	O
scalar	O
multiplication	O
by	O
is	O
a	O
linear	B-Architecture
homeomorphism	O
Since	O
and	O
it	O
follows	O
that	O
where	O
because	O
is	O
open	O
,	O
there	O
exists	O
some	O
which	O
satisfies	O
Define	O
by	O
which	O
is	O
a	O
homeomorphism	O
because	O
The	O
set	O
is	O
thus	O
an	O
open	O
subset	O
of	O
that	O
moreover	O
contains	O
If	O
then	O
since	O
is	O
convex	O
,	O
and	O
which	O
proves	O
that	O
Thus	O
is	O
an	O
open	O
subset	O
of	O
that	O
contains	O
the	O
origin	O
and	O
is	O
contained	O
in	O
Q.E.D.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
Hausdorff	O
if	O
and	O
only	O
if	O
is	O
a	O
closed	O
subset	O
of	O
or	O
equivalently	O
,	O
if	O
and	O
only	O
if	O
Because	O
is	O
a	O
vector	O
subspace	O
of	O
the	O
same	O
is	O
true	O
of	O
its	O
closure	O
which	O
is	O
referred	O
to	O
as	O
in	O
This	O
vector	O
space	O
satisfies	O
so	O
that	O
in	O
particular	O
,	O
every	O
neighborhood	O
of	O
the	O
origin	O
in	O
contains	O
the	O
vector	O
space	O
as	O
a	O
subset	O
.	O
</s>
<s>
Every	O
subset	O
of	O
also	O
carries	O
the	O
trivial	O
topology	O
and	O
so	O
is	O
itself	O
a	O
compact	O
,	O
and	O
thus	O
also	O
complete	B-Algorithm
,	O
subspace	O
(	O
see	O
footnote	O
for	O
a	O
proof	O
)	O
.Since	O
has	O
the	O
trivial	O
topology	O
,	O
so	O
does	O
each	O
of	O
its	O
subsets	O
,	O
which	O
makes	O
them	O
all	O
compact	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
a	O
subset	O
of	O
any	O
uniform	O
space	O
is	O
compact	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
and	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
image	O
if	O
under	O
the	O
canonical	O
quotient	O
map	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Closed	O
and	O
compact	O
setsCompact	O
and	O
totally	O
bounded	B-Algorithm
setsA	O
subset	O
of	O
a	O
TVS	O
is	O
compact	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
and	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Thus	O
,	O
in	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
,	O
a	O
closed	O
and	O
totally	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
compact	O
.	O
</s>
<s>
A	O
subset	O
of	O
a	O
TVS	O
is	O
totally	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
is	O
totally	O
bounded	B-Algorithm
,	O
if	O
and	O
only	O
if	O
its	O
image	O
under	O
the	O
canonical	O
quotient	O
map	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Every	O
relatively	O
compact	O
set	O
is	O
totally	O
bounded	B-Algorithm
and	O
the	O
closure	O
of	O
a	O
totally	O
bounded	B-Algorithm
set	I-Algorithm
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
image	O
of	O
a	O
totally	O
bounded	B-Algorithm
set	I-Algorithm
under	O
a	O
uniformly	O
continuous	O
map	O
(	O
such	O
as	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
for	O
instance	O
)	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
subset	O
of	O
a	O
TVS	O
such	O
that	O
every	O
sequence	O
in	O
has	O
a	O
cluster	O
point	O
in	O
then	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
compact	O
subset	O
of	O
a	O
TVS	O
and	O
is	O
an	O
open	O
subset	O
of	O
containing	O
then	O
there	O
exists	O
a	O
neighborhood	O
of	O
0	O
such	O
that	O
Closure	O
and	O
closed	O
setThe	O
closure	O
of	O
any	O
convex	O
(	O
respectively	O
,	O
any	O
balanced	O
,	O
any	O
absorbing	B-Algorithm
)	O
subset	O
of	O
any	O
TVS	O
has	O
this	O
same	O
property	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
closure	O
of	O
any	O
convex	O
,	O
balanced	O
,	O
and	O
absorbing	B-Algorithm
subset	O
is	O
a	O
barrel	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
sum	O
of	O
two	O
closed	O
subsets	O
may	O
fail	O
to	O
be	O
closed	O
(	O
see	O
this	O
footnoteIn	O
the	O
sum	O
of	O
the	O
-axis	O
and	O
the	O
graph	O
of	O
which	O
is	O
the	O
complement	O
of	O
the	O
-axis	O
,	O
is	O
open	O
in	O
In	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
is	O
a	O
countable	O
dense	O
subset	O
of	O
so	O
not	O
closed	O
in	O
for	O
examples	O
)	O
.	O
</s>
<s>
It	O
follows	O
that	O
for	O
every	O
neighborhood	O
of	O
the	O
origin	O
in	O
Closed	O
hullsIn	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
convex	O
hulls	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
are	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
balanced	O
hull	O
of	O
a	O
compact	O
(	O
respectively	O
,	O
totally	O
bounded	B-Algorithm
)	O
set	O
has	O
that	O
same	O
property	O
.	O
</s>
<s>
Then	O
is	O
a	O
Baire	O
space	O
if	O
and	O
only	O
if	O
has	O
no	O
balanced	O
absorbing	B-Algorithm
nowhere	O
dense	O
subset	O
.	O
</s>
<s>
Every	O
nonmeager	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
a	O
barrelled	O
space.Important	O
algebraic	O
facts	O
and	O
common	O
misconceptionsIf	O
then	O
;	O
if	O
is	O
convex	O
then	O
equality	O
holds	O
.	O
</s>
<s>
The	O
balanced	O
hull	O
of	O
a	O
compact	O
(	O
respectively	O
,	O
totally	O
bounded	B-Algorithm
,	O
open	O
)	O
set	O
has	O
that	O
same	O
property	O
.	O
</s>
<s>
The	O
(	O
Minkowski	O
)	O
sum	O
of	O
two	O
compact	O
(	O
respectively	O
,	O
bounded	B-Algorithm
,	O
balanced	O
,	O
convex	O
)	O
sets	O
has	O
that	O
same	O
property	O
.	O
</s>
<s>
And	O
the	O
convex	O
hull	O
of	O
a	O
bounded	B-Algorithm
set	I-Algorithm
need	O
be	O
bounded	B-Algorithm
.	O
</s>
<s>
So	O
for	O
instance	O
,	O
since	O
the	O
union	O
of	O
two	O
absorbing	B-Algorithm
sets	I-Algorithm
is	O
again	O
absorbing	B-Algorithm
,	O
the	O
cell	O
in	O
row	O
""	O
and	O
column	O
"	O
Absorbing	B-Algorithm
"	O
is	O
colored	O
green	O
.	O
</s>
<s>
But	O
since	O
the	O
arbitrary	O
intersection	O
of	O
absorbing	B-Algorithm
sets	I-Algorithm
need	O
not	O
be	O
absorbing	B-Algorithm
,	O
the	O
cell	O
in	O
row	O
"	O
Arbitrary	O
intersections	O
(	O
of	O
at	O
least	O
1	O
set	O
)	O
"	O
and	O
column	O
"	O
Absorbing	B-Algorithm
"	O
is	O
colored	O
red	O
.	O
</s>
