<s>
F	B-Algorithm
.	I-Algorithm
Riesz	I-Algorithm
's	I-Algorithm
theorem	I-Algorithm
(	O
named	O
after	O
Frigyes	O
Riesz	O
)	O
is	O
an	O
important	O
theorem	O
in	O
functional	B-Application
analysis	I-Application
that	O
states	O
that	O
a	O
Hausdorff	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
is	O
finite-dimensional	O
if	O
and	O
only	O
if	O
it	O
is	O
locally	O
compact	O
.	O
</s>
<s>
The	O
theorem	O
and	O
its	O
consequences	O
are	O
used	O
ubiquitously	O
in	O
functional	B-Application
analysis	I-Application
,	O
often	O
used	O
without	O
being	O
explicitly	O
mentioned	O
.	O
</s>
<s>
A	O
map	O
between	O
two	O
TVSs	O
is	O
called	O
a	O
TVS-isomorphism	O
or	O
an	O
isomorphism	O
in	O
the	O
category	O
of	O
TVSs	O
if	O
it	O
is	O
a	O
linear	B-Architecture
homeomorphism	O
.	O
</s>
<s>
a	O
linear	B-Architecture
bijection	B-Algorithm
)	O
between	O
two	O
finite-dimensional	O
Hausdorff	O
TVSs	O
is	O
a	O
TVS	O
isomorphism	O
.	O
</s>
<s>
Finite-dimensional	O
domain	O
:	O
A	O
linear	B-Architecture
map	I-Architecture
between	O
Hausdorff	O
TVSs	O
is	O
necessarily	O
continuous	O
.	O
</s>
<s>
In	O
particular	O
,	O
every	O
linear	B-Algorithm
functional	I-Algorithm
of	O
a	O
finite-dimensional	O
Hausdorff	O
TVS	O
is	O
continuous	O
.	O
</s>
<s>
Finite-dimensional	O
range	O
:	O
Any	O
continuous	O
surjective	B-Algorithm
linear	B-Architecture
map	I-Architecture
with	O
a	O
Hausdorff	O
finite-dimensional	O
range	O
is	O
an	O
open	O
map	O
and	O
thus	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
