<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
is	O
a	O
complete	O
compactly	O
generated	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
having	O
a	O
sequence	O
of	O
compact	O
sets	O
such	O
that	O
every	O
other	O
compact	O
set	O
is	O
contained	O
in	O
some	O
.	O
</s>
<s>
Brauner	B-Algorithm
spaces	I-Algorithm
are	O
named	O
after	O
,	O
who	O
began	O
their	O
study	O
.	O
</s>
<s>
All	O
Brauner	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
Fréchet	B-Algorithm
spaces	I-Algorithm
:	O
</s>
<s>
for	O
any	O
Fréchet	B-Algorithm
space	I-Algorithm
its	O
stereotype	O
dual	O
space	O
is	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
,	O
</s>
<s>
and	O
vice	O
versa	O
,	O
for	O
any	O
Brauner	B-Algorithm
space	I-Algorithm
its	O
stereotype	O
dual	O
space	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Special	O
cases	O
of	O
Brauner	B-Algorithm
spaces	I-Algorithm
are	O
Smith	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
-compact	O
locally	O
compact	O
topological	O
space	O
,	O
and	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
of	O
all	O
continuous	O
functions	O
on	O
(	O
with	O
values	O
in	O
or	O
)	O
,	O
endowed	O
with	O
the	O
usual	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
sets	O
in	O
.	O
</s>
<s>
The	O
dual	O
space	O
of	O
Radon	O
measures	O
with	O
compact	O
support	O
on	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
sets	O
in	O
is	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
smooth	O
manifold	O
,	O
and	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
of	O
all	O
smooth	O
functions	O
on	O
(	O
with	O
values	O
in	O
or	O
)	O
,	O
endowed	O
with	O
the	O
usual	O
topology	O
of	O
uniform	O
convergence	O
with	O
each	O
derivative	O
on	O
compact	O
sets	O
in	O
.	O
</s>
<s>
The	O
dual	O
space	O
of	O
distributions	O
with	O
compact	O
support	O
in	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
bounded	O
sets	O
in	O
is	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
Stein	O
manifold	O
and	O
the	O
Fréchet	B-Algorithm
space	I-Algorithm
of	O
all	O
holomorphic	O
functions	O
on	O
with	O
the	O
usual	O
topology	O
of	O
uniform	O
convergence	O
on	O
compact	O
sets	O
in	O
.	O
</s>
<s>
The	O
dual	O
space	O
of	O
analytic	O
functionals	O
on	O
with	O
the	O
topology	O
of	O
uniform	O
convergence	O
on	O
bounded	O
sets	O
in	O
is	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
polynomials	O
(	O
or	O
regular	O
functions	O
)	O
on	O
,	O
being	O
endowed	O
with	O
the	O
strongest	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
,	O
becomes	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Its	O
stereotype	O
dual	O
space	O
(	O
of	O
currents	O
on	O
)	O
is	O
a	O
Fréchet	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
space	O
of	O
all	O
holomorphic	O
functions	O
of	O
exponential	O
type	O
on	O
is	O
a	O
Brauner	B-Algorithm
space	I-Algorithm
with	O
respect	O
to	O
a	O
natural	O
topology	O
.	O
</s>
