<s>
In	O
mathematics	O
,	O
an	O
LB-space	B-Algorithm
,	O
also	O
written	O
(	O
LB	O
)	O
-space	O
,	O
is	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
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
Banach	O
spaces	O
.	O
</s>
<s>
This	O
means	O
that	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
Banach	O
space	O
.	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
LB-space	B-Algorithm
is	O
called	O
a	O
strict	O
LB-space	B-Algorithm
.	O
</s>
<s>
Schaefer	O
)	O
define	O
the	O
term	O
"	O
LB-space	B-Algorithm
"	O
to	O
mean	O
"	O
strict	O
LB-space	B-Algorithm
,	O
"	O
so	O
when	O
reading	O
mathematical	O
literature	O
,	O
its	O
recommended	O
to	O
always	O
check	O
how	O
LB-space	B-Algorithm
is	O
defined	O
.	O
</s>
<s>
A	O
strict	O
LB-space	B-Algorithm
is	O
complete	B-Algorithm
,	O
barrelled	B-Algorithm
,	O
and	O
bornological	B-Algorithm
(	O
and	O
thus	O
ultrabornological	B-Algorithm
)	O
.	O
</s>
<s>
If	O
is	O
a	O
locally	O
compact	O
topological	O
space	O
that	O
is	O
countable	O
at	O
infinity	O
(	O
that	O
is	O
,	O
it	O
is	O
equal	O
to	O
a	O
countable	O
union	O
of	O
compact	O
subspaces	O
)	O
then	O
the	O
space	O
of	O
all	O
continuous	O
,	O
complex-valued	O
functions	O
on	O
with	O
compact	O
support	O
is	O
a	O
strict	O
LB-space	B-Algorithm
.	O
</s>
<s>
With	O
this	O
topology	O
,	O
becomes	O
a	O
complete	B-Algorithm
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
sequential	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
that	O
is	O
a	O
Fréchet	O
–	O
Urysohn	O
space	O
.	O
</s>
<s>
This	O
makes	O
into	O
an	O
LB-space	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
a	O
bornological	B-Algorithm
LB-space	B-Algorithm
whose	O
strong	O
bidual	O
is	O
bornological	B-Algorithm
.	O
</s>
<s>
There	O
exists	O
an	O
LB-space	B-Algorithm
that	O
is	O
not	O
quasi-complete	B-Algorithm
.	O
</s>
