<s>
In	O
mathematics	O
,	O
an	O
LF-space	B-Algorithm
,	O
also	O
written	O
(	O
LF	O
)	O
-space	O
,	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
X	O
that	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
inductive	O
limit	O
of	O
a	O
countable	O
inductive	O
system	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
X	O
is	O
a	O
direct	O
limit	O
of	O
a	O
direct	O
system	O
in	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
and	O
each	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
name	O
LF	O
stands	O
for	O
Limit	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
If	O
each	O
of	O
the	O
bonding	O
maps	O
is	O
an	O
embedding	O
of	O
TVSs	O
then	O
the	O
LF-space	B-Algorithm
is	O
called	O
a	O
strict	B-Algorithm
LF-space	I-Algorithm
.	O
</s>
<s>
Schaefer	O
)	O
define	O
the	O
term	O
"	O
LF-space	B-Algorithm
"	O
to	O
mean	O
"	O
strict	B-Algorithm
LF-space	I-Algorithm
,	O
"	O
so	O
when	O
reading	O
mathematical	O
literature	O
,	O
it	O
is	O
recommended	O
to	O
always	O
check	O
how	O
LF-space	B-Algorithm
is	O
defined	O
.	O
</s>
<s>
is	O
either	O
the	O
category	O
of	O
topological	O
spaces	O
or	O
some	O
subcategory	O
of	O
the	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
;	O
</s>
<s>
In	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
,	O
the	O
topology	O
on	O
the	O
direct	O
limit	O
of	O
an	O
injective	O
directed	O
inductive	O
limit	O
of	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
can	O
be	O
described	O
by	O
specifying	O
that	O
an	O
absolutely	O
convex	O
subset	O
of	O
is	O
a	O
neighborhood	O
of	O
if	O
and	O
only	O
if	O
is	O
an	O
absolutely	O
convex	O
neighborhood	O
of	O
in	O
for	O
every	O
index	O
.	O
</s>
<s>
Direct	O
limits	O
of	O
directed	O
direct	O
systems	O
always	O
exist	O
in	O
the	O
categories	O
of	O
sets	O
,	O
topological	O
spaces	O
,	O
groups	O
,	O
and	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
.	O
</s>
<s>
surjective	B-Algorithm
,	O
bijective	B-Algorithm
,	O
homeomorphism	O
,	O
topological	O
embedding	O
,	O
quotient	O
map	O
)	O
then	O
so	O
is	O
every	O
.	O
</s>
<s>
Direct	O
limits	O
in	O
the	O
categories	O
of	O
topological	O
spaces	O
,	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
,	O
and	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
are	O
"	O
poorly	O
behaved	O
"	O
.	O
</s>
<s>
indexed	O
by	O
the	O
natural	O
numbers	O
)	O
of	O
locally	B-Algorithm
convex	I-Algorithm
nuclear	B-Algorithm
Fréchet	B-Algorithm
spaces	I-Algorithm
may	O
to	O
be	O
Hausdorff	O
(	O
in	O
which	O
case	O
the	O
direct	O
limit	O
does	O
not	O
exist	O
in	O
the	O
category	O
of	O
Hausdorff	O
TVSs	O
)	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
only	O
certain	O
"	O
well-behaved	O
"	O
direct	O
systems	O
are	O
usually	O
studied	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
Such	O
systems	O
include	O
LF-spaces	B-Algorithm
.	O
</s>
<s>
However	O
,	O
non-Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
inductive	O
limits	O
do	O
occur	O
in	O
natural	O
questions	O
of	O
analysis	O
.	O
</s>
<s>
In	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
spaces	I-Algorithm
,	O
the	O
topology	O
on	O
a	O
strict	O
inductive	O
limit	O
of	O
Fréchet	B-Algorithm
spaces	I-Algorithm
can	O
be	O
described	O
by	O
specifying	O
that	O
an	O
absolutely	O
convex	O
subset	O
is	O
a	O
neighborhood	O
of	O
if	O
and	O
only	O
if	O
is	O
an	O
absolutely	O
convex	O
neighborhood	O
of	O
in	O
for	O
every	O
.	O
</s>
<s>
An	O
inductive	O
limit	O
in	O
the	O
category	O
of	O
locally	B-Algorithm
convex	I-Algorithm
TVSs	O
of	O
a	O
family	O
of	O
bornological	B-Algorithm
(	O
resp	O
.	O
</s>
<s>
barrelled	B-Algorithm
,	O
quasi-barrelled	B-Algorithm
)	O
spaces	O
has	O
this	O
same	O
property	O
.	O
</s>
<s>
Every	O
LF-space	B-Algorithm
is	O
a	O
meager	O
subset	O
of	O
itself	O
.	O
</s>
<s>
The	O
strict	O
inductive	O
limit	O
of	O
a	O
sequence	O
of	O
complete	O
locally	B-Algorithm
convex	I-Algorithm
spaces	I-Algorithm
(	O
such	O
as	O
Fréchet	B-Algorithm
spaces	I-Algorithm
)	O
is	O
necessarily	O
complete	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
LF-space	B-Algorithm
is	O
complete	O
.	O
</s>
<s>
Every	O
LF-space	B-Algorithm
is	O
barrelled	B-Algorithm
and	O
bornological	B-Algorithm
,	O
which	O
together	O
with	O
completeness	O
implies	O
that	O
every	O
LF-space	B-Algorithm
is	O
ultrabornological	B-Algorithm
.	O
</s>
<s>
An	O
LF-space	B-Algorithm
that	O
is	O
the	O
inductive	O
limit	O
of	O
a	O
countable	O
sequence	O
of	O
separable	O
spaces	O
is	O
separable	O
.	O
</s>
<s>
LF	B-Algorithm
spaces	I-Algorithm
are	O
distinguished	B-Algorithm
and	O
their	O
strong	B-Algorithm
duals	I-Algorithm
are	O
bornological	B-Algorithm
and	O
barrelled	B-Algorithm
(	O
a	O
result	O
due	O
to	O
Alexander	O
Grothendieck	O
)	O
.	O
</s>
<s>
If	O
is	O
the	O
strict	O
inductive	O
limit	O
of	O
an	O
increasing	O
sequence	O
of	O
Fréchet	B-Algorithm
space	I-Algorithm
then	O
a	O
subset	O
of	O
is	O
bounded	O
in	O
if	O
and	O
only	O
if	O
there	O
exists	O
some	O
such	O
that	O
is	O
a	O
bounded	O
subset	O
of	O
.	O
</s>
<s>
A	O
linear	O
map	O
from	O
an	O
LF-space	B-Algorithm
into	O
another	O
TVS	O
is	O
continuous	O
if	O
and	O
only	O
if	O
it	O
is	O
sequentially	O
continuous	O
.	O
</s>
<s>
A	O
linear	O
map	O
from	O
an	O
LF-space	B-Algorithm
into	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
is	O
continuous	O
if	O
and	O
only	O
if	O
its	O
graph	O
is	O
closed	O
in	O
.	O
</s>
<s>
Every	O
bounded	O
linear	O
operator	O
from	O
an	O
LF-space	B-Algorithm
into	O
another	O
TVS	O
is	O
continuous	O
.	O
</s>
<s>
If	O
is	O
an	O
LF-space	B-Algorithm
defined	O
by	O
a	O
sequence	O
then	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
all	O
are	O
normable	O
.	O
</s>
<s>
Thus	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
an	O
LF-space	B-Algorithm
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
an	O
LB-space	B-Algorithm
.	O
</s>
<s>
A	O
typical	O
example	O
of	O
an	O
LF-space	B-Algorithm
is	O
,	O
,	O
the	O
space	O
of	O
all	O
infinitely	O
differentiable	O
functions	O
on	O
with	O
compact	O
support	O
.	O
</s>
<s>
The	O
LF-space	B-Algorithm
structure	O
is	O
obtained	O
by	O
considering	O
a	O
sequence	O
of	O
compact	O
sets	O
with	O
and	O
for	O
all	O
i	O
,	O
is	O
a	O
subset	O
of	O
the	O
interior	O
of	O
.	O
</s>
<s>
The	O
space	O
of	O
infinitely	O
differentiable	O
functions	O
on	O
with	O
compact	O
support	O
contained	O
in	O
has	O
a	O
natural	O
Fréchet	B-Algorithm
space	I-Algorithm
structure	O
and	O
inherits	O
its	O
LF-space	B-Algorithm
structure	O
as	O
described	O
above	O
.	O
</s>
<s>
The	O
LF-space	B-Algorithm
topology	O
does	O
not	O
depend	O
on	O
the	O
particular	O
sequence	O
of	O
compact	O
sets	O
.	O
</s>
<s>
With	O
this	O
LF-space	B-Algorithm
structure	O
,	O
is	O
known	O
as	O
the	O
space	O
of	O
test	O
functions	O
,	O
of	O
fundamental	O
importance	O
in	O
the	O
theory	O
of	O
distributions	O
.	O
</s>
<s>
Denote	O
the	O
resulting	O
LF-space	B-Algorithm
by	O
.	O
</s>
<s>
The	O
continuous	O
dual	O
space	O
of	O
is	O
equal	O
to	O
the	O
algebraic	O
dual	O
space	O
of	O
,	O
that	O
is	O
the	O
space	O
of	O
all	O
real	O
valued	O
sequences	O
and	O
the	O
weak	O
topology	O
on	O
is	O
equal	O
to	O
the	O
strong	B-Algorithm
topology	I-Algorithm
on	O
(	O
i.e.	O
</s>
<s>
Furthermore	O
,	O
the	O
canonical	O
map	O
of	O
into	O
the	O
continuous	O
dual	O
space	O
of	O
is	O
surjective	B-Algorithm
.	O
</s>
