<s>
In	O
the	O
field	O
of	O
functional	B-Application
analysis	I-Application
,	O
DF-spaces	B-Algorithm
,	O
also	O
written	O
(	O
DF	O
)	O
-spaces	O
are	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
having	O
a	O
property	O
that	O
is	O
shared	O
by	O
locally	B-Algorithm
convex	I-Algorithm
metrizable	B-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
DF-spaces	B-Algorithm
were	O
first	O
defined	O
by	O
Alexander	O
Grothendieck	O
and	O
studied	O
in	O
detail	O
by	O
him	O
in	O
.	O
</s>
<s>
Grothendieck	O
was	O
led	O
to	O
introduce	O
these	O
spaces	O
by	O
the	O
following	O
property	O
of	O
strong	B-Algorithm
duals	I-Algorithm
of	O
metrizable	B-Algorithm
spaces	O
:	O
If	O
is	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
and	O
is	O
a	O
sequence	O
of	O
convex	O
0-neighborhoods	O
in	O
such	O
that	O
absorbs	O
every	O
strongly	O
bounded	O
set	O
,	O
then	O
is	O
a	O
0-neighborhood	O
in	O
(	O
where	O
is	O
the	O
continuous	O
dual	O
space	O
of	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
)	O
.	O
</s>
<s>
is	O
a	O
countably	B-Algorithm
quasi-barrelled	I-Algorithm
space	I-Algorithm
(	O
i.e.	O
</s>
<s>
Let	O
be	O
a	O
DF-space	B-Algorithm
and	O
let	O
be	O
a	O
convex	O
balanced	O
subset	O
of	O
Then	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
if	O
and	O
only	O
if	O
for	O
every	O
convex	O
,	O
balanced	O
,	O
bounded	O
subset	O
is	O
a	O
neighborhood	O
of	O
the	O
origin	O
in	O
Consequently	O
,	O
a	O
linear	O
map	O
from	O
a	O
DF-space	B-Algorithm
into	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
continuous	O
if	O
its	O
restriction	O
to	O
each	O
bounded	O
subset	O
of	O
the	O
domain	O
is	O
continuous	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
DF-space	B-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
infinite-dimensional	O
Montel	B-Algorithm
DF-space	B-Algorithm
is	O
a	O
sequential	O
space	O
but	O
a	O
Fréchet	O
–	O
Urysohn	O
space	O
.	O
</s>
<s>
Suppose	O
is	O
either	O
a	O
DF-space	B-Algorithm
or	O
an	O
LM-space	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
sequential	O
space	O
then	O
it	O
is	O
either	O
metrizable	B-Algorithm
or	O
else	O
a	O
Montel	B-Algorithm
space	I-Algorithm
DF-space	B-Algorithm
.	O
</s>
<s>
Every	O
quasi-complete	B-Algorithm
DF-space	B-Algorithm
is	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
complete	B-Algorithm
nuclear	B-Algorithm
DF-space	B-Algorithm
then	O
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
is	O
a	O
DF-space	B-Algorithm
but	O
the	O
convers	O
is	O
in	O
general	O
not	O
true	O
(	O
the	O
converse	O
being	O
the	O
statement	O
that	O
every	O
DF-space	B-Algorithm
is	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
some	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
)	O
.	O
</s>
<s>
Every	O
normed	O
space	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
Every	O
Banach	O
space	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
Every	O
infrabarreled	B-Algorithm
space	I-Algorithm
possessing	O
a	O
fundamental	O
sequence	O
of	O
bounded	O
sets	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
Every	O
Hausdorff	O
quotient	O
of	O
a	O
DF-space	B-Algorithm
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
completion	B-Algorithm
of	O
a	O
DF-space	B-Algorithm
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
sum	O
of	O
a	O
sequence	O
of	O
DF-spaces	B-Algorithm
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
An	O
inductive	O
limit	O
of	O
a	O
sequence	O
of	O
DF-spaces	B-Algorithm
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
<	O
li>Suppose	O
that	O
and	O
are	O
DF-spaces	B-Algorithm
.	O
</s>
<s>
An	O
infinite	O
product	O
of	O
non-trivial	O
DF-spaces	B-Algorithm
(	O
i.e.	O
</s>
<s>
all	O
factors	O
have	O
non-0	O
dimension	O
)	O
is	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
A	O
closed	O
vector	O
subspace	O
of	O
a	O
DF-space	B-Algorithm
is	O
not	O
necessarily	O
a	O
DF-space	B-Algorithm
.	O
</s>
<s>
There	O
exist	O
complete	B-Algorithm
DF-spaces	B-Algorithm
that	O
are	O
not	O
TVS-isomorphic	O
to	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
There	O
exist	O
complete	B-Algorithm
DF-spaces	B-Algorithm
that	O
are	O
not	O
TVS-isomorphic	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
There	O
exist	O
DF-spaces	B-Algorithm
having	O
closed	O
vector	O
subspaces	O
that	O
are	O
not	O
DF-spaces	B-Algorithm
.	O
</s>
