<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
with	O
the	O
property	O
that	O
whenever	O
points	O
get	O
progressively	O
closer	O
to	O
each	O
other	O
,	O
then	O
there	O
exists	O
some	O
point	O
towards	O
which	O
they	O
all	O
get	O
closer	O
.	O
</s>
<s>
But	O
unlike	O
metric-completeness	O
,	O
TVS-completeness	O
does	O
not	O
depend	O
on	O
any	O
metric	O
and	O
is	O
defined	O
for	O
TVSs	O
,	O
including	O
those	O
that	O
are	O
not	O
metrizable	B-Algorithm
or	O
Hausdorff	O
.	O
</s>
<s>
Completeness	O
is	O
an	O
extremely	O
important	O
property	O
for	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
to	O
possess	O
.	O
</s>
<s>
The	O
notions	O
of	O
completeness	O
for	O
normed	O
spaces	O
and	O
metrizable	B-Algorithm
TVSs	O
,	O
which	O
are	O
commonly	O
defined	O
in	O
terms	O
of	O
completeness	O
of	O
a	O
particular	O
norm	O
or	O
metric	O
,	O
can	O
both	O
be	O
reduced	O
down	O
to	O
this	O
notion	O
of	O
TVS-completeness	O
a	O
notion	O
that	O
is	O
independent	O
of	O
any	O
particular	O
norm	O
or	O
metric	O
.	O
</s>
<s>
Prominent	O
examples	O
of	O
complete	O
TVSs	O
that	O
are	O
also	O
metrizable	B-Algorithm
include	O
all	O
F-spaces	B-Algorithm
and	O
consequently	O
also	O
all	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
Banach	O
spaces	O
,	O
and	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Prominent	O
examples	O
of	O
complete	B-Algorithm
TVS	I-Algorithm
that	O
are	O
(	O
typically	O
)	O
metrizable	B-Algorithm
include	O
strict	B-Algorithm
LF-spaces	I-Algorithm
such	O
as	O
the	O
space	B-Algorithm
of	I-Algorithm
test	I-Algorithm
functions	I-Algorithm
with	O
it	O
canonical	O
LF-topology	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
any	O
non-normable	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
as	O
well	O
as	O
many	O
other	O
polar	B-Algorithm
topologies	I-Algorithm
on	O
continuous	O
dual	O
space	O
or	O
other	O
topologies	B-Algorithm
on	I-Algorithm
spaces	I-Algorithm
of	I-Algorithm
linear	I-Algorithm
maps	I-Algorithm
.	O
</s>
<s>
Explicitly	O
,	O
a	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVS	O
)	O
is	O
complete	O
if	O
every	O
net	O
,	O
or	O
equivalently	O
,	O
every	O
filter	O
,	O
that	O
is	O
Cauchy	O
with	O
respect	O
to	O
the	O
space	O
's	O
necessarily	O
converges	O
to	O
some	O
point	O
.	O
</s>
<s>
Said	O
differently	O
,	O
a	O
TVS	O
is	O
complete	O
if	O
its	O
canonical	B-Algorithm
uniformity	I-Algorithm
is	O
a	O
complete	O
uniformity	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
even	O
if	O
it	O
is	O
not	O
metrizable	B-Algorithm
or	O
not	O
Hausdorff	O
,	O
has	O
a	O
,	O
which	O
by	O
definition	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
into	O
which	O
can	O
be	O
TVS-embedded	O
as	O
a	O
dense	O
vector	O
subspace	O
.	O
</s>
<s>
Moreover	O
,	O
every	O
Hausdorff	O
TVS	O
has	O
a	O
completion	B-Algorithm
,	O
which	O
is	O
necessarily	O
unique	O
up	O
to	O
TVS-isomorphism	O
.	O
</s>
<s>
This	O
section	O
summarizes	O
the	O
definition	O
of	O
a	O
complete	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
(	O
TVS	O
)	O
in	O
terms	O
of	O
both	O
nets	O
and	O
prefilters	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
a	O
commutative	O
topological	O
group	O
with	O
identity	O
under	O
addition	O
and	O
the	O
canonical	B-Algorithm
uniformity	I-Algorithm
of	O
a	O
TVS	O
is	O
defined	O
in	O
terms	O
of	O
subtraction	O
(	O
and	O
thus	O
addition	O
)	O
;	O
scalar	O
multiplication	O
is	O
not	O
involved	O
and	O
no	O
additional	O
structure	O
is	O
needed	O
.	O
</s>
<s>
The	O
same	O
canonical	B-Algorithm
uniformity	I-Algorithm
would	O
result	O
by	O
using	O
a	O
neighborhood	O
basis	O
of	O
the	O
origin	O
rather	O
the	O
filter	O
of	O
all	O
neighborhoods	O
of	O
the	O
origin	O
.	O
</s>
<s>
For	O
the	O
canonical	B-Algorithm
uniformity	I-Algorithm
on	O
these	O
definitions	O
reduce	O
down	O
to	O
those	O
given	O
below	O
.	O
</s>
<s>
A	O
series	O
is	O
called	O
a	O
(	O
respectively	O
,	O
a	O
)	O
if	O
the	O
sequence	O
of	O
partial	O
sums	O
is	O
a	O
Cauchy	O
sequence	O
(	O
respectively	O
,	O
a	O
convergent	B-Algorithm
sequence	I-Algorithm
)	O
.	O
</s>
<s>
In	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
,	O
every	O
Cauchy	O
series	O
is	O
necessarily	O
a	O
convergent	O
series	O
.	O
</s>
<s>
A	O
prefilter	O
on	O
an	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
a	O
Cauchy	O
prefilter	O
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
If	O
is	O
a	O
prefilter	O
on	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
and	O
if	O
then	O
in	O
if	O
and	O
only	O
if	O
and	O
is	O
Cauchy	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
called	O
a	O
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
are	O
satisfied	O
:	O
</s>
<s>
is	O
a	O
complete	O
uniform	O
space	O
when	O
it	O
is	O
endowed	O
with	O
its	O
canonical	B-Algorithm
uniformity	I-Algorithm
.	O
</s>
<s>
When	O
is	O
a	O
TVS	O
,	O
the	O
topology	O
induced	O
by	O
the	O
canonical	B-Algorithm
uniformity	I-Algorithm
is	O
equal	O
to	O
'	O
s	O
given	O
topology	O
(	O
so	O
convergence	O
in	O
this	O
induced	O
topology	O
is	O
just	O
the	O
usual	O
convergence	O
in	O
)	O
.	O
</s>
<s>
where	O
if	O
in	O
addition	O
is	O
pseudometrizable	B-Algorithm
or	O
metrizable	B-Algorithm
(	O
for	O
example	O
,	O
a	O
normed	O
space	O
)	O
then	O
this	O
list	O
can	O
be	O
extended	O
to	O
include	O
:	O
</s>
<s>
is	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
is	O
if	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
are	O
satisfied	O
:	O
</s>
<s>
is	O
a	O
sequentially	B-Algorithm
complete	I-Algorithm
subset	O
of	O
itself	O
.	O
</s>
<s>
The	O
existence	O
of	O
the	O
canonical	B-Algorithm
uniformity	I-Algorithm
was	O
demonstrated	O
above	O
by	O
defining	O
it	O
.	O
</s>
<s>
The	O
canonical	B-Algorithm
uniformity	I-Algorithm
on	O
any	O
TVS	O
is	O
translation-invariant	O
.	O
</s>
<s>
and	O
the	O
topology	O
induced	O
on	O
by	O
the	O
canonical	B-Algorithm
uniformity	I-Algorithm
is	O
the	O
same	O
as	O
the	O
topology	O
that	O
started	O
with	O
(	O
that	O
is	O
,	O
it	O
is	O
)	O
.	O
</s>
<s>
a	O
sequentially	B-Algorithm
complete	I-Algorithm
)	O
pseudometric	O
space	O
if	O
and	O
only	O
if	O
is	O
a	O
complete	O
(	O
resp	O
.	O
</s>
<s>
a	O
sequentially	B-Algorithm
complete	I-Algorithm
)	O
uniform	O
space	O
.	O
</s>
<s>
the	O
uniform	O
space	O
)	O
is	O
complete	O
if	O
and	O
only	O
if	O
it	O
is	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
The	O
canonical	B-Algorithm
uniformity	I-Algorithm
on	O
induced	O
by	O
the	O
pseudometric	O
is	O
a	O
complete	O
uniformity	O
.	O
</s>
<s>
Every	O
F-space	B-Algorithm
,	O
and	O
thus	O
also	O
every	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
Banach	O
space	O
,	O
and	O
Hilbert	O
space	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
every	O
F-space	B-Algorithm
is	O
a	O
Baire	O
space	O
but	O
there	O
are	O
normed	O
spaces	O
that	O
are	O
Baire	O
but	O
not	O
Banach	O
.	O
</s>
<s>
If	O
is	O
translation-invariant	O
,	O
then	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
if	O
and	O
only	O
if	O
is	O
a	O
complete	O
pseudometric	O
space	O
.	O
</s>
<s>
If	O
is	O
translation-invariant	O
,	O
then	O
may	O
be	O
possible	O
for	O
to	O
be	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
but	O
to	O
be	O
a	O
complete	O
pseudometric	O
space	O
(	O
see	O
this	O
footnote	O
for	O
an	O
example	O
)	O
.	O
</s>
<s>
Every	O
Banach	O
space	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
A	O
normed	O
space	O
is	O
a	O
Banach	O
space	O
(	O
that	O
is	O
,	O
its	O
canonical	O
norm-induced	O
metric	O
is	O
complete	O
)	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	O
as	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
A	O
completion	B-Algorithm
of	O
a	O
TVS	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
that	O
contains	O
a	O
dense	O
vector	O
subspace	O
that	O
is	O
TVS-isomorphic	O
to	O
In	O
other	O
words	O
,	O
it	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
into	O
which	O
can	O
be	O
TVS-embedded	O
as	O
a	O
dense	O
vector	O
subspace	O
.	O
</s>
<s>
Every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
has	O
a	O
completion	B-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
every	O
Hausdorff	O
TVS	O
has	O
a	O
completion	B-Algorithm
,	O
which	O
is	O
necessarily	O
unique	O
up	O
to	O
TVS-isomorphism	O
.	O
</s>
<s>
However	O
,	O
all	O
TVSs	O
,	O
even	O
those	O
that	O
are	O
Hausdorff	O
,	O
(	O
already	O
)	O
complete	O
,	O
and/or	O
metrizable	B-Algorithm
have	O
infinitely	O
many	O
non-Hausdorff	O
completions	O
that	O
are	O
TVS-isomorphic	O
to	O
one	O
another	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
vector	O
space	O
consisting	O
of	O
scalar-valued	O
simple	O
functions	O
for	O
which	O
(	O
where	O
this	O
seminorm	O
is	O
defined	O
in	O
the	O
usual	O
way	O
in	O
terms	O
of	O
Lebesgue	O
integration	O
)	O
becomes	O
a	O
seminormed	O
space	O
when	O
endowed	O
with	O
this	O
seminorm	O
,	O
which	O
in	O
turn	O
makes	O
it	O
into	O
both	O
a	O
pseudometric	O
space	O
and	O
a	O
non-Hausdorff	O
non-complete	O
TVS	O
;	O
any	O
completion	B-Algorithm
of	O
this	O
space	O
is	O
a	O
non-Hausdorff	O
complete	O
seminormed	O
space	O
that	O
when	O
quotiented	O
by	O
the	O
closure	O
of	O
its	O
origin	O
(	O
so	O
as	O
to	O
obtain	O
a	O
Hausdorff	O
TVS	O
)	O
results	O
in	O
(	O
a	O
space	O
linearly	B-Architecture
isometrically-isomorphic	O
to	O
)	O
the	O
usual	O
complete	O
Hausdorff	O
-space	O
(	O
endowed	O
with	O
the	O
usual	O
complete	O
norm	O
)	O
.	O
</s>
<s>
As	O
another	O
example	O
demonstrating	O
the	O
usefulness	O
of	O
completions	O
,	O
the	O
completions	O
of	O
topological	O
tensor	O
products	O
,	O
such	O
as	O
projective	B-Algorithm
tensor	I-Algorithm
products	I-Algorithm
or	O
injective	B-Algorithm
tensor	I-Algorithm
products	I-Algorithm
,	O
of	O
the	O
Banach	O
space	O
with	O
a	O
complete	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
results	O
in	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
that	O
is	O
TVS-isomorphic	O
to	O
a	O
"	O
generalized	O
"	O
-space	O
consisting	O
-valued	O
functions	O
on	O
(	O
where	O
this	O
"	O
generalized	O
"	O
TVS	O
is	O
defined	O
analogously	O
to	O
original	O
space	O
of	O
scalar-valued	O
functions	O
on	O
)	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
completion	B-Algorithm
of	O
the	O
injective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
the	O
space	O
of	O
scalar-valued	O
-test	O
functions	O
with	O
such	O
a	O
TVS	O
is	O
TVS-isomorphic	O
to	O
the	O
analogously	O
defined	O
TVS	O
of	O
-valued	O
test	O
functions	O
.	O
</s>
<s>
As	O
the	O
example	O
below	O
shows	O
,	O
regardless	O
of	O
whether	O
or	O
not	O
a	O
space	O
is	O
Hausdorff	O
or	O
already	O
complete	O
,	O
every	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
has	O
infinitely	O
many	O
non-isomorphic	O
completions	O
.	O
</s>
<s>
However	O
,	O
every	O
Hausdorff	O
TVS	O
has	O
a	O
completion	B-Algorithm
that	O
is	O
unique	O
up	O
to	O
TVS-isomorphism	O
.	O
</s>
<s>
Let	O
denote	O
any	O
complete	B-Algorithm
TVS	I-Algorithm
and	O
let	O
denote	O
any	O
TVS	O
endowed	O
with	O
the	O
indiscrete	O
topology	O
,	O
which	O
recall	O
makes	O
into	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Thus	O
by	O
definition	O
of	O
"	O
completion	B-Algorithm
,	O
"	O
is	O
a	O
completion	B-Algorithm
of	O
(	O
it	O
does	O
n't	O
matter	O
that	O
is	O
already	O
complete	O
)	O
.	O
</s>
<s>
Every	O
Hausdorff	O
TVS	O
has	O
a	O
completion	B-Algorithm
that	O
is	O
unique	O
up	O
to	O
TVS-isomorphism	O
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	B-Algorithm
TVS	I-Algorithm
then	O
a	O
Hausdorff	O
completion	B-Algorithm
of	O
can	O
be	O
constructed	O
using	O
equivalence	O
classes	O
of	O
Cauchy	O
sequences	O
instead	O
of	O
minimal	O
Cauchy	O
filters	O
.	O
</s>
<s>
This	O
subsection	O
details	O
how	O
every	O
non-Hausdorff	O
TVS	O
can	O
be	O
TVS-embedded	O
onto	O
a	O
dense	O
vector	O
subspace	O
of	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
The	O
proof	O
that	O
every	O
Hausdorff	O
TVS	O
has	O
a	O
Hausdorff	O
completion	B-Algorithm
is	O
widely	O
available	O
and	O
so	O
this	O
fact	O
will	O
be	O
used	O
(	O
without	O
proof	O
)	O
to	O
show	O
that	O
every	O
non-Hausdorff	O
TVS	O
also	O
has	O
a	O
completion	B-Algorithm
.	O
</s>
<s>
Let	O
denote	O
any	O
topological	B-Algorithm
complement	I-Algorithm
of	O
in	O
which	O
is	O
necessarily	O
a	O
Hausdorff	O
TVS	O
(	O
since	O
it	O
is	O
TVS-isomorphic	O
to	O
the	O
quotient	O
TVS	O
)	O
.	O
</s>
<s>
is	O
a	O
well-defined	O
TVS-embedding	O
of	O
onto	O
a	O
dense	O
vector	O
subspace	O
of	O
the	O
complete	B-Algorithm
TVS	I-Algorithm
where	O
moreover	O
,	O
</s>
<s>
Let	O
denote	O
the	O
on	O
the	O
continuous	O
dual	O
space	O
which	O
by	O
definition	O
consists	O
of	O
all	O
equicontinuous	O
weak-*	O
closed	O
and	O
weak-*	O
bounded	B-Algorithm
absolutely	O
convex	O
subsets	O
of	O
(	O
which	O
are	O
necessarily	O
weak-*	O
compact	O
subsets	O
of	O
)	O
.	O
</s>
<s>
If	O
a	O
TVS	O
has	O
any	O
of	O
the	O
following	O
properties	O
then	O
so	O
does	O
its	O
completion	B-Algorithm
:	O
</s>
<s>
Moreover	O
,	O
if	O
is	O
a	O
normed	O
space	O
,	O
then	O
the	O
completion	B-Algorithm
can	O
be	O
chosen	O
to	O
be	O
a	O
Banach	O
space	O
such	O
that	O
the	O
TVS-embedding	O
of	O
into	O
is	O
an	O
isometry	O
.	O
</s>
<s>
If	O
is	O
a	O
Hausdorff	O
TVS	O
,	O
then	O
the	O
continuous	O
dual	O
space	O
of	O
is	O
identical	O
to	O
the	O
continuous	O
dual	O
space	O
of	O
the	O
completion	B-Algorithm
of	O
The	O
completion	B-Algorithm
of	O
a	O
locally	B-Algorithm
convex	I-Algorithm
bornological	B-Algorithm
space	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
DF-spaces	B-Algorithm
then	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
,	O
as	O
well	O
as	O
its	O
completion	B-Algorithm
,	O
of	O
these	O
spaces	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
completion	B-Algorithm
of	O
the	O
projective	B-Algorithm
tensor	I-Algorithm
product	I-Algorithm
of	O
two	O
nuclear	B-Algorithm
spaces	I-Algorithm
is	O
nuclear	B-Algorithm
.	O
</s>
<s>
The	O
completion	B-Algorithm
of	O
a	O
nuclear	B-Algorithm
space	I-Algorithm
is	O
TVS-isomorphic	O
with	O
a	O
projective	O
limit	O
of	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Let	O
be	O
a	O
completion	B-Algorithm
of	O
Let	O
denote	O
the	O
vector	O
space	O
of	O
continuous	O
linear	B-Architecture
operators	I-Architecture
and	O
let	O
denote	O
the	O
map	O
that	O
sends	O
every	O
to	O
its	O
unique	O
continuous	O
linear	O
extension	O
on	O
Then	O
is	O
a	O
(	O
surjective	O
)	O
vector	O
space	O
isomorphism	O
.	O
</s>
<s>
Thus	O
a	O
complete	O
seminormable	O
locally	B-Algorithm
convex	I-Algorithm
and	O
locally	O
compact	O
TVS	O
need	O
not	O
be	O
finite-dimensional	O
if	O
it	O
is	O
not	O
Hausdorff	O
.	O
</s>
<s>
sequentially	B-Algorithm
complete	I-Algorithm
,	O
quasi-complete	B-Algorithm
)	O
TVSs	O
has	O
that	O
same	O
property	O
.	O
</s>
<s>
A	O
product	O
of	O
Hausdorff	O
completions	O
of	O
a	O
family	O
of	O
(	O
Hausdorff	O
)	O
TVSs	O
is	O
a	O
Hausdorff	O
completion	B-Algorithm
of	O
their	O
product	O
TVS	O
.	O
</s>
<s>
sequentially	B-Algorithm
complete	I-Algorithm
,	O
quasi-complete	B-Algorithm
)	O
TVSs	O
has	O
that	O
same	O
property	O
.	O
</s>
<s>
A	O
projective	O
limit	O
of	O
Hausdorff	O
completions	O
of	O
an	O
inverse	O
system	O
of	O
(	O
Hausdorff	O
)	O
TVSs	O
is	O
a	O
Hausdorff	O
completion	B-Algorithm
of	O
their	O
projective	O
limit	O
.	O
</s>
<s>
If	O
is	O
a	O
closed	O
vector	O
subspace	O
of	O
a	O
complete	O
pseudometrizable	B-Algorithm
TVS	I-Algorithm
then	O
the	O
quotient	O
space	O
is	O
complete	O
.	O
</s>
<s>
Suppose	O
is	O
a	O
vector	O
subspace	O
of	O
a	O
metrizable	B-Algorithm
TVS	I-Algorithm
If	O
the	O
quotient	O
space	O
is	O
complete	O
then	O
so	O
is	O
However	O
,	O
there	O
exists	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
having	O
a	O
closed	O
vector	O
subspace	O
such	O
that	O
the	O
quotient	O
TVS	O
is	O
complete	O
.	O
</s>
<s>
Every	O
F-space	B-Algorithm
,	O
Fréchet	B-Algorithm
space	I-Algorithm
,	O
Banach	O
space	O
,	O
and	O
Hilbert	O
space	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
Strict	B-Algorithm
LF-spaces	I-Algorithm
and	O
strict	O
LB-spaces	B-Algorithm
are	O
complete	O
.	O
</s>
<s>
The	O
Schwartz	B-Algorithm
space	I-Algorithm
of	O
smooth	O
functions	O
is	O
complete	O
.	O
</s>
<s>
Suppose	O
that	O
and	O
are	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
and	O
that	O
the	O
space	O
of	O
continuous	O
linear	B-Architecture
maps	I-Architecture
is	O
endowed	O
with	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	I-Algorithm
bounded	I-Algorithm
subsets	I-Algorithm
of	O
If	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
and	O
if	O
is	O
complete	O
then	O
is	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
bornological	B-Algorithm
space	I-Algorithm
is	O
complete	O
.	O
</s>
<s>
However	O
,	O
it	O
need	O
not	O
be	O
bornological	B-Algorithm
.	O
</s>
<s>
Every	O
quasi-complete	B-Algorithm
DF-space	B-Algorithm
is	O
complete	O
.	O
</s>
<s>
Let	O
and	O
be	O
Hausdorff	O
TVS	O
topologies	O
on	O
a	O
vector	O
space	O
such	O
that	O
If	O
there	O
exists	O
a	O
prefilter	O
such	O
that	O
is	O
a	O
neighborhood	O
basis	O
at	O
the	O
origin	O
for	O
and	O
such	O
that	O
every	O
is	O
a	O
complete	O
subset	O
of	O
then	O
is	O
a	O
complete	B-Algorithm
TVS	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
complete	B-Algorithm
TVS	I-Algorithm
is	O
quasi-complete	B-Algorithm
space	I-Algorithm
and	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
sequentially	B-Algorithm
complete	I-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVS	O
that	O
is	O
not	O
quasi-complete	B-Algorithm
.	O
</s>
<s>
Every	O
complete	O
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
and	O
a	O
Baire	O
space	O
(	O
and	O
thus	O
non-meager	O
)	O
.	O
</s>
<s>
The	O
dimension	O
of	O
a	O
complete	O
metrizable	B-Algorithm
TVS	I-Algorithm
is	O
either	O
finite	O
or	O
uncountable	O
.	O
</s>
<s>
A	O
Cauchy	O
sequence	O
in	O
a	O
Hausdorff	O
TVS	O
A	O
Cauchy	O
sequence	O
in	O
a	O
Hausdorff	O
TVS	O
when	O
considered	O
as	O
a	O
set	O
,	O
is	O
not	O
necessarily	O
relatively	O
compact	O
(	O
that	O
is	O
,	O
its	O
closure	O
in	O
is	O
not	O
necessarily	O
compact	O
)	O
although	O
it	O
is	O
precompact	O
(	O
that	O
is	O
,	O
its	O
closure	O
in	O
the	O
completion	B-Algorithm
of	O
is	O
compact	O
)	O
.	O
</s>
<s>
Every	O
Cauchy	O
sequence	O
is	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
but	O
this	O
is	O
not	O
necessarily	O
true	O
of	O
Cauchy	O
net	O
.	O
</s>
<s>
This	O
net	O
is	O
a	O
Cauchy	O
net	O
in	O
because	O
it	O
converges	O
to	O
the	O
origin	O
,	O
but	O
the	O
set	O
is	O
not	O
a	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
(	O
because	O
does	O
not	O
have	O
the	O
trivial	O
topology	O
)	O
.	O
</s>
<s>
Suppose	O
be	O
a	O
continuous	O
linear	B-Architecture
operator	I-Architecture
between	O
two	O
Hausdorff	O
TVSs	O
.	O
</s>
<s>
If	O
is	O
a	O
dense	O
vector	O
subspace	O
of	O
and	O
if	O
the	O
restriction	O
to	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
then	O
is	O
also	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
So	O
if	O
and	O
are	O
Hausdorff	O
completions	O
of	O
and	O
respectively	O
,	O
and	O
if	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
,	O
then	O
'	O
s	O
unique	O
continuous	O
linear	O
extension	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
Suppose	O
and	O
are	O
Hausdorff	O
TVSs	O
,	O
is	O
a	O
dense	O
vector	O
subspace	O
of	O
and	O
is	O
a	O
dense	O
vector	O
subspaces	O
of	O
If	O
are	O
and	O
are	O
topologically	O
isomorphic	O
additive	O
subgroups	O
via	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
then	O
the	O
same	O
is	O
true	O
of	O
and	O
via	O
the	O
unique	O
uniformly	O
continuous	O
extension	O
of	O
(	O
which	O
is	O
also	O
a	O
homeomorphism	O
)	O
.	O
</s>
<s>
Every	O
complete	O
subset	O
of	O
a	O
TVS	O
is	O
sequentially	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
Closed	O
subsets	O
of	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
are	O
complete	O
;	O
however	O
,	O
if	O
a	O
TVS	O
is	O
not	O
complete	O
then	O
is	O
a	O
closed	O
subset	O
of	O
that	O
is	O
not	O
complete	O
.	O
</s>
<s>
If	O
is	O
a	O
non-normable	O
Fréchet	B-Algorithm
space	I-Algorithm
on	O
which	O
there	O
exists	O
a	O
continuous	O
norm	O
then	O
contains	O
a	O
closed	O
vector	O
subspace	O
that	O
has	O
no	O
topological	B-Algorithm
complement	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
separable	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
and	O
let	O
be	O
its	O
completion	B-Algorithm
.	O
</s>
<s>
A	O
subset	O
of	O
a	O
TVS	O
(	O
assumed	O
to	O
be	O
Hausdorff	O
or	O
complete	O
)	O
is	O
compact	O
if	O
and	O
only	O
if	O
it	O
is	O
complete	O
and	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Thus	O
a	O
closed	O
and	O
totally	O
bounded	B-Algorithm
subset	I-Algorithm
of	O
a	O
complete	B-Algorithm
TVS	I-Algorithm
is	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	O
locally	B-Algorithm
convex	I-Algorithm
Hausdorff	O
TVS	O
,	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>
However	O
,	O
like	O
in	O
all	O
complete	O
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>
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	O
,	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>
Every	O
complete	O
totally	O
bounded	B-Algorithm
set	I-Algorithm
is	O
relatively	O
compact	O
.	O
</s>
<s>
If	O
is	O
any	O
TVS	O
then	O
the	O
quotient	O
map	O
is	O
a	O
closed	O
map	O
and	O
thus	O
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
its	O
image	O
under	O
the	O
canonical	O
quotient	O
map	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
Thus	O
is	O
totally	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
any	O
TVS	O
,	O
the	O
closure	O
of	O
a	O
totally	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
again	O
totally	O
bounded	B-Algorithm
.	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>
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>
A	O
subset	O
of	O
a	O
Hausdorff	O
TVS	O
is	O
totally	O
bounded	B-Algorithm
if	O
and	O
only	O
if	O
every	O
ultrafilter	O
on	O
is	O
Cauchy	O
,	O
which	O
happens	O
if	O
and	O
only	O
if	O
it	O
is	O
pre-compact	O
(	O
that	O
is	O
,	O
its	O
closure	O
in	O
the	O
completion	B-Algorithm
of	O
is	O
compact	O
)	O
.	O
</s>
<s>
Every	O
relatively	O
compact	O
subset	O
of	O
a	O
Hausdorff	O
TVS	O
is	O
totally	O
bounded	B-Algorithm
.	O
</s>
<s>
In	O
a	O
complete	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
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
resp	O
.	O
</s>
