<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
Smith	B-Algorithm
space	I-Algorithm
is	O
a	O
complete	O
compactly	O
generated	O
locally	B-Algorithm
convex	I-Algorithm
topological	I-Algorithm
vector	I-Algorithm
space	I-Algorithm
having	O
a	O
universal	O
compact	O
set	O
,	O
i.e.	O
</s>
<s>
,	O
who	O
introduced	O
them	O
as	O
duals	O
to	O
Banach	O
spaces	O
in	O
some	O
versions	O
of	O
duality	O
theory	O
for	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
All	O
Smith	B-Algorithm
spaces	I-Algorithm
are	O
stereotype	O
and	O
are	O
in	O
the	O
stereotype	O
duality	O
relations	O
with	O
Banach	O
spaces	O
:	O
</s>
<s>
for	O
any	O
Banach	O
space	O
its	O
stereotype	O
dual	O
space	O
is	O
a	O
Smith	B-Algorithm
space	I-Algorithm
,	O
</s>
<s>
and	O
vice	O
versa	O
,	O
for	O
any	O
Smith	B-Algorithm
space	I-Algorithm
its	O
stereotype	O
dual	O
space	O
is	O
a	O
Banach	O
space	O
.	O
</s>
<s>
Smith	B-Algorithm
spaces	I-Algorithm
are	O
special	O
cases	O
of	O
Brauner	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
As	O
follows	O
from	O
the	O
duality	O
theorems	O
,	O
for	O
any	O
Banach	O
space	O
its	O
stereotype	O
dual	O
space	O
is	O
a	O
Smith	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
polar	B-Algorithm
of	O
the	O
unit	O
ball	O
in	O
is	O
the	O
universal	O
compact	O
set	O
in	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
for	O
infinite	O
dimensional	O
the	O
space	O
is	O
not	O
barreled	B-Algorithm
(	O
and	O
even	O
is	O
not	O
a	O
Mackey	B-Algorithm
space	I-Algorithm
if	O
is	O
reflexive	O
as	O
a	O
Banach	O
space	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
convex	O
balanced	O
compact	O
set	O
in	O
a	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
,	O
then	O
its	O
linear	O
span	O
possesses	O
a	O
unique	O
structure	O
of	O
a	O
Smith	B-Algorithm
space	I-Algorithm
with	O
as	O
the	O
universal	O
compact	O
set	O
(	O
and	O
with	O
the	O
same	O
topology	O
on	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
(	O
Hausdorff	O
)	O
compact	O
topological	O
space	O
,	O
and	O
the	O
Banach	O
space	O
of	O
continuous	O
functions	O
on	O
(	O
with	O
the	O
usual	O
sup-norm	O
)	O
,	O
then	O
the	O
stereotype	O
dual	O
space	O
(	O
of	O
Radon	O
measures	O
on	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
sets	O
in	O
)	O
is	O
a	O
Smith	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
A	O
Banach	O
space	O
is	O
a	O
Smith	B-Algorithm
space	I-Algorithm
if	O
and	O
only	O
if	O
is	O
finite-dimensional	O
.	O
</s>
