<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
(	O
LCTVS	B-Algorithm
)	O
or	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
are	O
examples	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVS	O
)	O
that	O
generalize	O
normed	O
spaces	O
.	O
</s>
<s>
They	O
can	O
be	O
defined	O
as	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
whose	O
topology	O
is	O
generated	O
by	O
translations	O
of	O
balanced	O
,	O
absorbent	B-Algorithm
,	O
convex	O
sets	O
.	O
</s>
<s>
Although	O
in	O
general	O
such	O
spaces	O
are	O
not	O
necessarily	O
normable	O
,	O
the	O
existence	O
of	O
a	O
convex	O
local	O
base	O
for	O
the	O
zero	O
vector	O
is	O
strong	O
enough	O
for	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
to	O
hold	O
,	O
yielding	O
a	O
sufficiently	O
rich	O
theory	O
of	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
.	O
</s>
<s>
Fréchet	B-Algorithm
spaces	I-Algorithm
are	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
that	O
are	O
completely	O
metrizable	O
(	O
with	O
a	O
choice	O
of	O
complete	B-Algorithm
metric	O
)	O
.	O
</s>
<s>
They	O
are	O
generalizations	O
of	O
Banach	O
spaces	O
,	O
which	O
are	O
complete	B-Algorithm
vector	O
spaces	O
with	O
respect	O
to	O
a	O
metric	O
generated	O
by	O
a	O
norm	O
.	O
</s>
<s>
After	O
the	O
notion	O
of	O
a	O
general	O
topological	O
space	O
was	O
defined	O
by	O
Felix	O
Hausdorff	O
in	O
1914	O
,	O
although	O
locally	B-Algorithm
convex	I-Algorithm
topologies	I-Algorithm
were	O
implicitly	O
used	O
by	O
some	O
mathematicians	O
,	O
up	O
to	O
1934	O
only	O
John	O
von	O
Neumann	O
would	O
seem	O
to	O
have	O
explicitly	O
defined	O
the	O
weak	O
topology	O
on	O
Hilbert	O
spaces	O
and	O
strong	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
on	O
operators	O
on	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Finally	O
,	O
in	O
1935	O
von	O
Neumann	O
introduced	O
the	O
general	O
definition	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
(	O
called	O
a	O
convex	O
space	O
by	O
him	O
)	O
.	O
</s>
<s>
A	O
notable	O
example	O
of	O
a	O
result	O
which	O
had	O
to	O
wait	O
for	O
the	O
development	O
and	O
dissemination	O
of	O
general	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
(	O
amongst	O
other	O
notions	O
and	O
results	O
,	O
like	O
nets	O
,	O
the	O
product	O
topology	O
and	O
Tychonoff	O
's	O
theorem	O
)	O
to	O
be	O
proven	O
in	O
its	O
full	O
generality	O
,	O
is	O
the	O
Banach	B-Algorithm
–	I-Algorithm
Alaoglu	I-Algorithm
theorem	I-Algorithm
which	O
Stefan	O
Banach	O
first	O
established	O
in	O
1932	O
by	O
an	O
elementary	O
diagonal	O
argument	O
for	O
the	O
case	O
of	O
separable	O
normed	O
spaces	O
(	O
in	O
which	O
case	O
the	O
unit	O
ball	O
of	O
the	O
dual	O
is	O
metrizable	O
)	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
defined	O
either	O
in	O
terms	O
of	O
convex	O
sets	O
,	O
or	O
equivalently	O
in	O
terms	O
of	O
seminorms	O
.	O
</s>
<s>
Absorbent	B-Algorithm
or	O
absorbing	B-Algorithm
if	O
for	O
every	O
there	O
exists	O
such	O
that	O
for	O
all	O
satisfying	O
The	O
set	O
can	O
be	O
scaled	O
out	O
by	O
any	O
"	O
large	O
"	O
value	O
to	O
absorb	O
every	O
point	O
in	O
the	O
space	O
.	O
</s>
<s>
In	O
any	O
TVS	O
,	O
every	O
neighborhood	O
of	O
the	O
origin	O
is	O
absorbent	B-Algorithm
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
called	O
locally	B-Algorithm
convex	I-Algorithm
if	O
the	O
origin	O
has	O
a	O
neighborhood	O
basis	O
(	O
that	O
is	O
,	O
a	O
local	O
base	O
)	O
consisting	O
of	O
convex	O
sets	O
.	O
</s>
<s>
In	O
fact	O
,	O
every	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
has	O
a	O
neighborhood	O
basis	O
of	O
the	O
origin	O
consisting	O
of	O
sets	O
(	O
that	O
is	O
,	O
disks	O
)	O
,	O
where	O
this	O
neighborhood	O
basis	O
can	O
further	O
be	O
chosen	O
to	O
also	O
consist	O
entirely	O
of	O
open	O
sets	O
or	O
entirely	O
of	O
closed	O
sets	O
.	O
</s>
<s>
Every	O
TVS	O
has	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
consisting	O
of	O
balanced	O
sets	O
but	O
only	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
has	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
consisting	O
of	O
sets	O
that	O
are	O
both	O
balanced	O
convex	O
.	O
</s>
<s>
It	O
is	O
possible	O
for	O
a	O
TVS	O
to	O
have	O
neighborhoods	O
of	O
the	O
origin	O
that	O
are	O
convex	O
and	O
yet	O
be	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Because	O
translation	O
is	O
(	O
by	O
definition	O
of	O
"	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
"	O
)	O
continuous	O
,	O
all	O
translations	O
are	O
homeomorphisms	O
,	O
so	O
every	O
base	O
for	O
the	O
neighborhoods	O
of	O
the	O
origin	O
can	O
be	O
translated	O
to	O
a	O
base	O
for	O
the	O
neighborhoods	O
of	O
any	O
given	O
vector	O
.	O
</s>
<s>
A	O
family	O
of	O
seminorms	O
on	O
the	O
vector	O
space	O
induces	O
a	O
canonical	O
vector	O
space	O
topology	O
on	O
,	O
called	O
the	O
initial	O
topology	O
induced	O
by	O
the	O
seminorms	O
,	O
making	O
it	O
into	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
.	O
</s>
<s>
It	O
is	O
possible	O
for	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
a	O
space	O
to	O
be	O
induced	O
by	O
a	O
family	O
of	O
norms	O
but	O
for	O
to	O
be	O
normable	O
(	O
that	O
is	O
,	O
to	O
have	O
its	O
topology	O
be	O
induced	O
by	O
a	O
single	O
norm	O
)	O
.	O
</s>
<s>
Intersections	O
of	O
finitely	O
many	O
such	O
sets	O
are	O
then	O
also	O
convex	O
,	O
and	O
since	O
the	O
collection	O
of	O
all	O
such	O
finite	O
intersections	O
is	O
a	O
basis	O
at	O
the	O
origin	O
it	O
follows	O
that	O
the	O
topology	O
is	O
locally	B-Algorithm
convex	I-Algorithm
in	O
the	O
sense	O
of	O
the	O
definition	O
given	O
above	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
if	O
is	O
a	O
collection	O
of	O
continuous	O
seminorms	O
on	O
,	O
then	O
is	O
called	O
a	O
base	O
of	O
continuous	O
seminorms	O
if	O
it	O
is	O
a	O
base	O
of	O
seminorms	O
for	O
the	O
collection	O
of	O
continuous	O
seminorms	O
on	O
.	O
</s>
<s>
If	O
is	O
a	O
base	O
of	O
continuous	O
seminorms	O
for	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
then	O
the	O
family	O
of	O
all	O
sets	O
of	O
the	O
form	O
as	O
varies	O
over	O
and	O
varies	O
over	O
the	O
positive	O
real	O
numbers	O
,	O
is	O
a	O
of	O
neighborhoods	O
of	O
the	O
origin	O
in	O
(	O
not	O
just	O
a	O
subbasis	O
,	O
so	O
there	O
is	O
no	O
need	O
to	O
take	O
finite	O
intersections	O
of	O
such	O
sets	O
)	O
.	O
</s>
<s>
The	O
following	O
theorem	O
implies	O
that	O
if	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
then	O
the	O
topology	O
of	O
can	O
be	O
a	O
defined	O
by	O
a	O
family	O
of	O
continuous	O
on	O
(	O
a	O
norm	O
is	O
a	O
seminorm	O
where	O
implies	O
)	O
if	O
and	O
only	O
if	O
there	O
exists	O
continuous	O
on	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
sum	O
of	O
a	O
norm	O
and	O
a	O
seminorm	O
is	O
a	O
norm	O
so	O
if	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
defined	O
by	O
some	O
family	O
of	O
seminorms	O
(	O
each	O
of	O
which	O
is	O
necessarily	O
continuous	O
)	O
then	O
the	O
family	O
of	O
(	O
also	O
continuous	O
)	O
norms	O
obtained	O
by	O
adding	O
some	O
given	O
continuous	O
norm	O
to	O
each	O
element	O
,	O
will	O
necessarily	O
be	O
a	O
family	O
of	O
norms	O
that	O
defines	O
this	O
same	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
If	O
there	O
exists	O
a	O
continuous	O
norm	O
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
then	O
is	O
necessarily	O
Hausdorff	O
but	O
the	O
converse	O
is	O
not	O
in	O
general	O
true	O
(	O
not	O
even	O
for	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
or	O
Fréchet	B-Algorithm
spaces	I-Algorithm
)	O
.	O
</s>
<s>
Suppose	O
that	O
the	O
topology	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
induced	O
by	O
a	O
family	O
of	O
continuous	O
seminorms	O
on	O
.	O
</s>
<s>
From	O
this	O
definition	O
it	O
follows	O
that	O
is	O
a	O
seminorm	O
if	O
is	O
balanced	O
and	O
convex	O
(	O
it	O
is	O
also	O
absorbent	B-Algorithm
by	O
assumption	O
)	O
.	O
</s>
<s>
form	O
a	O
base	O
of	O
convex	O
absorbent	B-Algorithm
balanced	O
sets	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
pseudometrizable	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	B-Algorithm
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	B-Algorithm
subset	O
is	O
also	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
then	O
the	O
absorbing	B-Algorithm
disk	O
will	O
also	O
be	O
bounded	B-Algorithm
,	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>
A	O
family	O
of	O
seminorms	O
is	O
called	O
total	O
or	O
separated	O
or	O
is	O
said	O
to	O
separate	O
points	O
if	O
whenever	O
holds	O
for	O
every	O
then	O
is	O
necessarily	O
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
Hausdorff	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
separated	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
A	O
pseudometric	O
is	O
a	O
generalization	O
of	O
a	O
metric	O
which	O
does	O
not	O
satisfy	O
the	O
condition	O
that	O
only	O
when	O
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
pseudometrizable	B-Algorithm
,	O
meaning	O
that	O
its	O
topology	O
arises	O
from	O
a	O
pseudometric	O
,	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
countable	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
If	O
furthermore	O
the	O
space	O
is	O
complete	B-Algorithm
,	O
the	O
space	O
is	O
called	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
As	O
with	O
any	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
,	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
also	O
a	O
uniform	O
space	O
.	O
</s>
<s>
Thus	O
one	O
may	O
speak	O
of	O
uniform	O
continuity	O
,	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
and	O
Cauchy	O
sequences	O
.	O
</s>
<s>
A	O
Cauchy	O
net	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
net	O
such	O
that	O
for	O
every	O
and	O
every	O
seminorm	O
there	O
exists	O
some	O
index	O
such	O
that	O
for	O
all	O
indices	O
In	O
other	O
words	O
,	O
the	O
net	O
must	O
be	O
Cauchy	O
in	O
all	O
the	O
seminorms	O
simultaneously	O
.	O
</s>
<s>
The	O
definition	O
of	O
completeness	O
is	O
given	O
here	O
in	O
terms	O
of	O
nets	O
instead	O
of	O
the	O
more	O
familiar	O
sequences	O
because	O
unlike	O
Fréchet	B-Algorithm
spaces	I-Algorithm
which	O
are	O
metrizable	O
,	O
general	O
spaces	O
may	O
be	O
defined	O
by	O
an	O
uncountable	O
family	O
of	O
pseudometrics	O
.	O
</s>
<s>
A	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
complete	B-Algorithm
if	O
and	O
only	O
if	O
every	O
Cauchy	O
net	O
converges	O
.	O
</s>
<s>
Any	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
a	O
finite	O
family	O
of	O
seminorms	O
is	O
seminormable	O
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
open	O
sets	O
,	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
seminormable	O
if	O
and	O
only	O
if	O
the	O
origin	O
has	O
a	O
bounded	B-Algorithm
neighborhood	O
.	O
</s>
<s>
Say	O
that	O
a	O
vector	O
subspace	O
of	O
has	O
the	O
extension	O
property	O
if	O
any	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
on	O
can	O
be	O
extended	O
to	O
a	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
on	O
.	O
</s>
<s>
The	O
Hahn-Banach	O
theorem	O
guarantees	O
that	O
every	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
has	O
the	O
HBEP	O
.	O
</s>
<s>
For	O
complete	B-Algorithm
metrizable	O
TVSs	O
there	O
is	O
a	O
converse	O
:	O
</s>
<s>
If	O
a	O
vector	O
space	O
has	O
uncountable	O
dimension	O
and	O
if	O
we	O
endow	O
it	O
with	O
the	O
finest	O
vector	O
topology	O
then	O
this	O
is	O
a	O
TVS	O
with	O
the	O
HBEP	O
that	O
is	O
neither	O
locally	B-Algorithm
convex	I-Algorithm
or	O
metrizable	O
.	O
</s>
<s>
Every	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
homeomorphic	O
to	O
a	O
vector	O
subspace	O
of	O
a	O
product	O
of	O
Banach	O
spaces	O
.	O
</s>
<s>
The	O
Anderson	O
–	O
Kadec	O
theorem	O
states	O
that	O
every	O
infinite	O
–	O
dimensional	O
separable	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
homeomorphic	O
to	O
the	O
product	O
space	O
of	O
countably	O
many	O
copies	O
of	O
(	O
this	O
homeomorphism	O
need	O
not	O
be	O
a	O
linear	B-Architecture
map	I-Architecture
)	O
.	O
</s>
<s>
The	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
two	O
convex	O
sets	O
is	O
convex	O
;	O
furthermore	O
,	O
the	O
scalar	O
multiple	O
of	O
a	O
convex	O
set	O
is	O
again	O
convex	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
TVS	O
(	O
not	O
necessarily	O
locally	B-Algorithm
convex	I-Algorithm
or	O
Hausdorff	O
)	O
over	O
the	O
real	O
or	O
complex	O
numbers	O
.	O
</s>
<s>
If	O
is	O
convex	O
and	O
then	O
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
belongs	O
to	O
the	O
closure	O
of	O
and	O
belongs	O
to	O
the	O
interior	O
of	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
</s>
<s>
The	O
closure	O
of	O
a	O
convex	O
subset	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
space	O
is	O
the	O
same	O
for	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
TVS	O
topologies	O
on	O
that	O
are	O
compatible	O
with	O
duality	B-Algorithm
between	O
and	O
its	O
continuous	O
dual	O
space	O
.	O
</s>
<s>
In	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
the	O
convex	O
hull	O
and	O
the	O
disked	O
hull	O
of	O
a	O
totally	O
bounded	B-Algorithm
set	I-Algorithm
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
a	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
the	O
convex	O
hull	O
and	O
the	O
disked	O
hull	O
of	O
a	O
compact	O
set	O
are	O
both	O
compact	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
is	O
a	O
compact	O
subset	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
convex	O
hull	O
(	O
respectively	O
,	O
the	O
disked	O
hull	O
)	O
is	O
compact	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
In	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>
In	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
the	O
closed	O
convex	O
hull	O
of	O
a	O
compact	O
set	O
is	O
compact	O
.	O
</s>
<s>
In	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
any	O
linear	O
combination	O
of	O
totally	O
bounded	B-Algorithm
sets	I-Algorithm
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
However	O
,	O
like	O
in	O
all	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
,	O
the	O
convex	O
hull	O
of	O
this	O
compact	O
subset	O
is	O
compact	O
.	O
</s>
<s>
The	O
vector	O
subspace	O
is	O
a	O
pre-Hilbert	O
space	O
when	O
endowed	O
with	O
the	O
substructure	O
that	O
the	O
Hilbert	O
space	O
induces	O
on	O
it	O
but	O
is	O
not	O
complete	B-Algorithm
and	O
(	O
since	O
)	O
.	O
</s>
<s>
The	O
closed	O
convex	O
hull	O
of	O
in	O
(	O
here	O
,	O
"	O
closed	O
"	O
means	O
with	O
respect	O
to	O
and	O
not	O
to	O
as	O
before	O
)	O
is	O
equal	O
to	O
which	O
is	O
not	O
compact	O
(	O
because	O
it	O
is	O
not	O
a	O
complete	B-Algorithm
subset	O
)	O
.	O
</s>
<s>
This	O
shows	O
that	O
in	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
that	O
is	O
not	O
complete	B-Algorithm
,	O
the	O
closed	O
convex	O
hull	O
of	O
compact	O
subset	O
might	O
to	O
be	O
compact	O
(	O
although	O
it	O
will	O
be	O
precompact/totally	O
bounded	B-Algorithm
)	O
.	O
</s>
<s>
In	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
the	O
closed	O
convex	O
hull	O
of	O
compact	O
subset	O
is	O
not	O
necessarily	O
compact	O
although	O
it	O
is	O
a	O
precompact	O
(	O
also	O
called	O
"	O
totally	O
bounded	B-Algorithm
"	O
)	O
subset	O
,	O
which	O
means	O
that	O
its	O
closure	O
,	O
of	O
will	O
be	O
compact	O
(	O
here	O
so	O
that	O
if	O
and	O
only	O
if	O
is	O
complete	B-Algorithm
)	O
;	O
that	O
is	O
to	O
say	O
,	O
will	O
be	O
compact	O
.	O
</s>
<s>
So	O
for	O
example	O
,	O
the	O
closed	O
convex	O
hull	O
of	O
a	O
compact	O
subset	O
of	O
of	O
a	O
pre-Hilbert	O
space	O
is	O
always	O
a	O
precompact	O
subset	O
of	O
and	O
so	O
the	O
closure	O
of	O
in	O
any	O
Hilbert	O
space	O
containing	O
(	O
such	O
as	O
the	O
Hausdorff	O
completion	B-Algorithm
of	O
for	O
instance	O
)	O
will	O
be	O
compact	O
(	O
this	O
is	O
the	O
case	O
in	O
the	O
previous	O
example	O
above	O
)	O
.	O
</s>
<s>
In	O
a	O
quasi-complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVS	O
,	O
the	O
closure	O
of	O
the	O
convex	O
hull	O
of	O
a	O
compact	O
subset	O
is	O
again	O
compact	O
.	O
</s>
<s>
In	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
,	O
the	O
convex	O
hull	O
of	O
a	O
precompact	O
set	O
is	O
again	O
precompact	O
.	O
</s>
<s>
Consequently	O
,	O
in	O
a	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
the	O
closed	O
convex	O
hull	O
of	O
a	O
compact	O
subset	O
is	O
again	O
compact	O
.	O
</s>
<s>
In	O
any	O
non-Hausdorff	O
TVS	O
,	O
there	O
exist	O
subsets	O
that	O
are	O
compact	O
(	O
and	O
thus	O
complete	B-Algorithm
)	O
but	O
closed	O
.	O
</s>
<s>
The	O
bipolar	O
theorem	O
states	O
that	O
the	O
bipolar	O
(	O
that	O
is	O
,	O
the	O
polar	B-Algorithm
of	O
the	O
polar	B-Algorithm
)	O
of	O
a	O
subset	O
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
TVS	O
is	O
equal	O
to	O
the	O
closed	O
convex	O
balanced	O
hull	O
of	O
that	O
set	O
.	O
</s>
<s>
Any	O
vector	O
space	O
endowed	O
with	O
the	O
trivial	O
topology	O
(	O
also	O
called	O
the	O
indiscrete	O
topology	O
)	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
(	O
and	O
of	O
course	O
,	O
it	O
is	O
the	O
coarsest	O
such	O
topology	O
)	O
.	O
</s>
<s>
The	O
indiscrete	O
topology	O
makes	O
any	O
vector	O
space	O
into	O
a	O
complete	B-Algorithm
pseudometrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
This	O
follows	O
from	O
the	O
fact	O
that	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
a	O
connected	O
space	O
.	O
</s>
<s>
Every	O
linear	B-Architecture
map	I-Architecture
from	O
into	O
another	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
linear	B-Algorithm
functional	I-Algorithm
on	O
is	O
continuous	O
and	O
every	O
vector	O
subspace	O
of	O
is	O
closed	O
in	O
;	O
</s>
<s>
therefore	O
,	O
if	O
is	O
infinite	O
dimensional	O
then	O
is	O
not	O
pseudometrizable	B-Algorithm
(	O
and	O
thus	O
not	O
metrizable	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
is	O
the	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
on	O
with	O
the	O
property	O
that	O
any	O
linear	B-Architecture
map	I-Architecture
from	O
it	O
into	O
any	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
continuous	O
.	O
</s>
<s>
The	O
space	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
normed	O
space	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
and	O
much	O
of	O
the	O
theory	O
of	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
generalizes	O
parts	O
of	O
the	O
theory	O
of	O
normed	O
spaces	O
.	O
</s>
<s>
Every	O
Banach	O
space	O
is	O
a	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
in	O
particular	O
,	O
the	O
spaces	O
with	O
are	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
every	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
A	O
Fréchet	B-Algorithm
space	I-Algorithm
can	O
be	O
defined	O
as	O
a	O
complete	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
a	O
separated	O
countable	O
family	O
of	O
seminorms	O
.	O
</s>
<s>
is	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
The	O
countable	O
family	O
of	O
seminorms	O
is	O
complete	B-Algorithm
and	O
separable	O
,	O
so	O
this	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
which	O
is	O
not	O
normable	O
.	O
</s>
<s>
Given	O
any	O
vector	O
space	O
and	O
a	O
collection	O
of	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
it	O
,	O
can	O
be	O
made	O
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
by	O
giving	O
it	O
the	O
weakest	O
topology	O
making	O
all	O
linear	B-Algorithm
functionals	I-Algorithm
in	O
continuous	O
.	O
</s>
<s>
The	O
family	O
of	O
seminorms	O
defined	O
by	O
is	O
separated	O
,	O
and	O
countable	O
,	O
and	O
the	O
space	O
is	O
complete	B-Algorithm
,	O
so	O
this	O
metrizable	O
space	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
It	O
is	O
known	O
as	O
the	O
Schwartz	B-Algorithm
space	I-Algorithm
,	O
or	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
of	O
rapid	O
decrease	O
,	O
and	O
its	O
dual	O
space	O
is	O
the	O
space	O
of	O
tempered	O
distributions	O
.	O
</s>
<s>
A	O
more	O
detailed	O
construction	O
is	O
needed	O
for	O
the	O
topology	O
of	O
this	O
space	O
because	O
the	O
space	O
is	O
not	O
complete	B-Algorithm
in	O
the	O
uniform	O
norm	O
.	O
</s>
<s>
The	O
topology	O
on	O
is	O
defined	O
as	O
follows	O
:	O
for	O
any	O
fixed	O
compact	O
set	O
the	O
space	O
of	O
functions	O
with	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
with	O
countable	O
family	O
of	O
seminorms	O
(	O
these	O
are	O
actually	O
norms	O
,	O
and	O
the	O
completion	B-Algorithm
of	O
the	O
space	O
with	O
the	O
norm	O
is	O
a	O
Banach	O
space	O
)	O
.	O
</s>
<s>
Such	O
a	O
limit	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
is	O
known	O
as	O
an	O
LF	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
More	O
concretely	O
,	O
is	O
the	O
union	O
of	O
all	O
the	O
with	O
the	O
strongest	O
topology	O
which	O
makes	O
each	O
inclusion	B-Algorithm
map	I-Algorithm
continuous	O
.	O
</s>
<s>
This	O
space	O
is	O
locally	B-Algorithm
convex	I-Algorithm
and	O
complete	B-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
not	O
metrizable	O
,	O
and	O
so	O
it	O
is	O
not	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
More	O
abstractly	O
,	O
given	O
a	O
topological	O
space	O
the	O
space	O
of	O
continuous	O
(	O
not	O
necessarily	O
bounded	B-Algorithm
)	O
functions	O
on	O
can	O
be	O
given	O
the	O
topology	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
on	O
compact	O
sets	O
.	O
</s>
<s>
Many	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
are	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
The	O
spaces	O
for	O
are	O
equipped	O
with	O
the	O
F-norm	B-Algorithm
They	O
are	O
not	O
locally	B-Algorithm
convex	I-Algorithm
,	O
since	O
the	O
only	O
convex	O
neighborhood	O
of	O
zero	O
is	O
the	O
whole	O
space	O
.	O
</s>
<s>
More	O
generally	O
the	O
spaces	O
with	O
an	O
atomless	O
,	O
finite	O
measure	O
and	O
are	O
not	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Both	O
examples	O
have	O
the	O
property	O
that	O
any	O
continuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
to	O
the	O
real	O
numbers	O
is	O
In	O
particular	O
,	O
their	O
dual	O
space	O
is	O
trivial	O
,	O
that	O
is	O
,	O
it	O
contains	O
only	O
the	O
zero	O
functional	O
.	O
</s>
<s>
The	O
sequence	B-Algorithm
space	I-Algorithm
is	O
not	O
locally	B-Algorithm
convex	I-Algorithm
.	O
</s>
<s>
Because	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
are	O
topological	O
spaces	O
as	O
well	O
as	O
vector	O
spaces	O
,	O
the	O
natural	O
functions	O
to	O
consider	O
between	O
two	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
are	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Using	O
the	O
seminorms	O
,	O
a	O
necessary	O
and	O
sufficient	O
criterion	O
for	O
the	O
continuity	O
of	O
a	O
linear	B-Architecture
map	I-Architecture
can	O
be	O
given	O
that	O
closely	O
resembles	O
the	O
more	O
familiar	O
boundedness	B-Algorithm
condition	O
found	O
for	O
Banach	O
spaces	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
each	O
seminorm	O
of	O
the	O
range	O
of	O
is	O
bounded	B-Algorithm
above	O
by	O
some	O
finite	O
sum	O
of	O
seminorms	O
in	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
class	O
of	O
all	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
forms	O
a	O
category	O
with	O
continuous	B-Algorithm
linear	I-Algorithm
maps	I-Algorithm
as	O
morphisms	O
.	O
</s>
<s>
Let	O
be	O
an	O
integer	O
,	O
be	O
TVSs	O
(	O
not	O
necessarily	O
locally	B-Algorithm
convex	I-Algorithm
)	O
,	O
let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
whose	O
topology	O
is	O
determined	O
by	O
a	O
family	O
of	O
continuous	O
seminorms	O
,	O
and	O
let	O
be	O
a	O
multilinear	O
operator	O
that	O
is	O
linear	O
in	O
each	O
of	O
its	O
coordinates	O
.	O
</s>
<s>
For	O
every	O
there	O
exists	O
some	O
neighborhood	O
of	O
the	O
origin	O
in	O
on	O
which	O
is	O
bounded	B-Algorithm
.	O
</s>
