<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
the	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
the	O
continuous	O
dual	O
space	O
of	O
equipped	O
with	O
the	O
strong	O
(	O
dual	O
)	O
topology	O
or	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
where	O
this	O
topology	O
is	O
denoted	O
by	O
or	O
The	O
coarsest	O
polar	O
topology	O
is	O
called	O
weak	O
topology	O
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
plays	O
such	O
an	O
important	O
role	O
in	O
modern	O
functional	B-Application
analysis	I-Application
,	O
that	O
the	O
continuous	O
dual	O
space	O
is	O
usually	O
assumed	O
to	O
have	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
unless	O
indicated	O
otherwise	O
.	O
</s>
<s>
To	O
emphasize	O
that	O
the	O
continuous	O
dual	O
space	O
,	O
has	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
,	O
or	O
may	O
be	O
written	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
the	O
usual	O
notion	O
of	O
bounded	B-Algorithm
subsets	I-Algorithm
when	O
is	O
given	O
the	O
weak	O
topology	O
induced	O
by	O
which	O
is	O
a	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
now	O
proceeds	O
as	O
in	O
the	O
case	O
of	O
a	O
TVS	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
when	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
the	O
on	O
the	O
(	O
continuous	O
)	O
dual	O
space	O
(	O
that	O
is	O
,	O
on	O
the	O
space	O
of	O
all	O
continuous	O
linear	O
functionals	O
)	O
is	O
defined	O
as	O
the	O
strong	B-Algorithm
topology	I-Algorithm
and	O
it	O
coincides	O
with	O
the	O
topology	B-Algorithm
of	I-Algorithm
uniform	I-Algorithm
convergence	I-Algorithm
on	O
bounded	B-Algorithm
sets	I-Algorithm
in	O
i.e.	O
</s>
<s>
Let	O
be	O
any	O
fundamental	O
system	O
of	O
bounded	B-Algorithm
sets	I-Algorithm
of	O
;	O
</s>
<s>
that	O
is	O
,	O
is	O
a	O
family	O
of	O
bounded	B-Algorithm
subsets	I-Algorithm
of	O
such	O
that	O
every	O
bounded	B-Algorithm
subset	O
of	O
is	O
a	O
subset	O
of	O
some	O
;	O
</s>
<s>
A	O
basis	O
of	O
closed	O
neighborhoods	O
of	O
the	O
origin	O
in	O
is	O
given	O
by	O
the	O
polars	B-Algorithm
:	O
</s>
<s>
This	O
is	O
a	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
that	O
is	O
given	O
by	O
the	O
set	O
of	O
seminorms	O
on	O
:	O
</s>
<s>
the	O
topology	O
that	O
this	O
norm	O
induces	O
on	O
is	O
identical	O
to	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
The	O
bidual	O
or	O
second	O
dual	O
of	O
a	O
TVS	O
often	O
denoted	O
by	O
is	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
:	O
</s>
<s>
Unless	O
indicated	O
otherwise	O
,	O
the	O
vector	O
space	O
is	O
usually	O
assumed	O
to	O
be	O
endowed	O
with	O
the	O
strong	B-Algorithm
dual	I-Algorithm
topology	I-Algorithm
induced	O
on	O
it	O
by	O
in	O
which	O
case	O
it	O
is	O
called	O
the	O
strong	O
bidual	O
of	O
;	O
that	O
is	O
,	O
</s>
<s>
Let	O
be	O
a	O
locally	B-Algorithm
convex	I-Algorithm
TVS	O
.	O
</s>
<s>
Every	O
weakly	O
bounded	B-Algorithm
subset	O
of	O
is	O
strongly	O
bounded	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
metrizable	B-Algorithm
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
the	O
strong	B-Algorithm
dual	I-Algorithm
of	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
an	O
infrabarreled	B-Algorithm
space	I-Algorithm
,	O
if	O
and	O
only	O
if	O
it	O
is	O
a	O
barreled	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
bornological	B-Algorithm
space	I-Algorithm
(	O
e.g.	O
</s>
<s>
metrizable	B-Algorithm
or	O
LF-space	B-Algorithm
)	O
then	O
is	O
complete	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
normed	O
vector	O
space	O
,	O
then	O
its	O
(	O
continuous	O
)	O
dual	O
space	O
with	O
the	O
strong	B-Algorithm
topology	I-Algorithm
coincides	O
with	O
the	O
Banach	O
dual	O
space	O
;	O
that	O
is	O
,	O
with	O
the	O
space	O
with	O
the	O
topology	O
induced	O
by	O
the	O
operator	O
norm	O
.	O
</s>
