<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
Schwartz	O
spaces	O
are	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVS	O
)	O
whose	O
neighborhoods	O
of	O
the	O
origin	O
have	O
a	O
property	O
similar	O
to	O
the	O
definition	O
of	O
totally	O
bounded	O
subsets	O
.	O
</s>
<s>
A	O
Hausdorff	O
locally	B-Algorithm
convex	I-Algorithm
space	I-Algorithm
with	O
continuous	O
dual	O
,	O
is	O
called	O
a	O
Schwartz	O
space	O
if	O
it	O
satisfies	O
any	O
of	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
Every	O
quasi-complete	B-Algorithm
Schwartz	O
space	O
is	O
a	O
semi-Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Every	O
Fréchet	B-Algorithm
Schwartz	O
space	O
is	O
a	O
Montel	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
strong	B-Algorithm
dual	I-Algorithm
space	I-Algorithm
of	O
a	O
complete	B-Algorithm
Schwartz	O
space	O
is	O
an	O
ultrabornological	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
locally	B-Algorithm
convex	I-Algorithm
strict	O
inductive	O
limit	O
of	O
any	O
countable	O
sequence	O
of	O
Schwartz	O
spaces	O
(	O
with	O
each	O
space	O
TVS-embedded	O
in	O
the	O
next	O
space	O
)	O
is	O
again	O
a	O
Schwartz	O
space	O
.	O
</s>
<s>
There	O
exist	O
Fréchet	B-Algorithm
spaces	I-Algorithm
that	O
are	O
not	O
Schwartz	O
spaces	O
and	O
there	O
exist	O
Schwartz	O
spaces	O
that	O
are	O
not	O
Montel	B-Algorithm
spaces	I-Algorithm
.	O
</s>
