<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
Montel	B-Algorithm
space	I-Algorithm
,	O
named	O
after	O
Paul	O
Montel	O
,	O
is	O
any	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
in	O
which	O
an	O
analog	O
of	O
Montel	O
's	O
theorem	O
holds	O
.	O
</s>
<s>
Specifically	O
,	O
a	O
Montel	B-Algorithm
space	I-Algorithm
is	O
a	O
barrelled	B-Algorithm
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
in	O
which	O
every	O
closed	O
and	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
compact	O
.	O
</s>
<s>
A	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
has	O
the	O
if	O
every	O
closed	O
and	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
compact	O
.	O
</s>
<s>
A	O
is	O
a	O
barrelled	B-Algorithm
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
with	O
the	O
Heine	O
–	O
Borel	O
property	O
.	O
</s>
<s>
Equivalently	O
,	O
it	O
is	O
an	O
infrabarrelled	B-Algorithm
semi-Montel	O
space	O
where	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
is	O
called	O
a	O
or	O
if	O
every	O
bounded	B-Algorithm
subset	I-Algorithm
is	O
relatively	O
compact	O
.	O
</s>
<s>
A	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>
A	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
that	O
is	O
also	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
separable	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
each	O
weak-*	O
convergent	O
sequence	O
in	O
its	O
continuous	O
dual	O
is	O
strongly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
direct	O
sum	O
of	O
any	O
family	O
of	O
semi-Montel	O
spaces	O
is	O
again	O
a	O
semi-Montel	O
space	O
.	O
</s>
<s>
Montel	B-Algorithm
spaces	I-Algorithm
)	O
is	O
again	O
a	O
semi-Montel	O
space	O
(	O
resp	O
.	O
</s>
<s>
a	O
Montel	B-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
Montel	B-Algorithm
space	I-Algorithm
is	O
Montel	O
.	O
</s>
<s>
A	O
barrelled	B-Algorithm
quasi-complete	B-Algorithm
nuclear	B-Algorithm
space	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
product	O
and	O
locally	B-Algorithm
convex	I-Algorithm
direct	O
sum	O
of	O
a	O
family	O
of	O
Montel	B-Algorithm
spaces	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strict	O
inductive	O
limit	O
of	O
a	O
sequence	O
of	O
Montel	B-Algorithm
spaces	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
contrast	O
,	O
closed	O
subspaces	O
and	O
separated	O
quotients	O
of	O
Montel	B-Algorithm
spaces	I-Algorithm
are	O
in	O
general	O
not	O
even	O
reflexive	O
.	O
</s>
<s>
Every	O
Fréchet	B-Algorithm
Schwartz	B-Algorithm
space	I-Algorithm
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Montel	B-Algorithm
spaces	I-Algorithm
are	O
paracompact	O
and	O
normal	O
.	O
</s>
<s>
Semi-Montel	O
spaces	O
are	O
quasi-complete	B-Algorithm
and	O
semi-reflexive	O
while	O
Montel	B-Algorithm
spaces	I-Algorithm
are	O
reflexive	O
.	O
</s>
<s>
No	O
infinite-dimensional	O
Banach	O
space	O
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
This	O
is	O
because	O
a	O
Banach	O
space	O
cannot	O
satisfy	O
the	O
Heine	O
–	O
Borel	O
property	O
:	O
the	O
closed	O
unit	O
ball	O
is	O
closed	O
and	O
bounded	B-Algorithm
,	O
but	O
not	O
compact	O
.	O
</s>
<s>
Fréchet	B-Algorithm
Montel	B-Algorithm
spaces	I-Algorithm
are	O
separable	O
and	O
have	O
a	O
bornological	B-Algorithm
strong	B-Algorithm
dual	I-Algorithm
.	O
</s>
<s>
A	O
metrizable	O
Montel	B-Algorithm
space	I-Algorithm
is	O
separable	O
.	O
</s>
<s>
Fréchet	B-Algorithm
–	O
Montel	B-Algorithm
spaces	I-Algorithm
are	O
distinguished	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Many	O
Montel	B-Algorithm
spaces	I-Algorithm
of	O
contemporary	O
interest	O
arise	O
as	O
spaces	O
of	O
test	O
functions	O
for	O
a	O
space	O
of	O
distributions	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
space	O
of	O
compactly	O
supported	O
functions	O
in	O
an	O
open	O
set	O
with	O
the	O
final	O
topology	O
of	O
the	O
family	O
of	O
inclusions	O
as	O
ranges	O
over	O
all	O
compact	O
subsets	O
of	O
The	O
Schwartz	B-Algorithm
space	I-Algorithm
is	O
also	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
infinite-dimensional	O
normed	O
space	O
is	O
a	O
barrelled	B-Algorithm
space	I-Algorithm
that	O
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
infinite-dimensional	O
Banach	O
space	O
is	O
not	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
There	O
exist	O
Montel	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
separable	O
and	O
there	O
exist	O
Montel	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
complete	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
Montel	B-Algorithm
spaces	I-Algorithm
having	O
closed	O
vector	O
subspaces	O
that	O
are	O
Montel	B-Algorithm
spaces	I-Algorithm
.	O
</s>
