<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
or	O
simply	O
homomorphism	O
(	O
if	O
no	O
confusion	O
will	O
arise	O
)	O
is	O
the	O
analog	O
of	O
homomorphisms	O
for	O
the	O
category	O
of	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
(	O
TVSs	O
)	O
.	O
</s>
<s>
This	O
concept	O
is	O
of	O
considerable	O
importance	O
in	O
functional	B-Application
analysis	I-Application
and	O
the	O
famous	O
open	O
mapping	O
theorem	O
gives	O
a	O
sufficient	O
condition	O
for	O
a	O
continuous	O
linear	B-Architecture
map	I-Architecture
between	O
Fréchet	B-Algorithm
spaces	I-Algorithm
to	O
be	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
This	O
concept	O
is	O
of	O
considerable	O
importance	O
in	O
functional	B-Application
analysis	I-Application
and	O
the	O
famous	O
open	O
mapping	O
theorem	O
gives	O
a	O
sufficient	O
condition	O
for	O
a	O
continuous	O
linear	B-Architecture
map	I-Architecture
between	O
Fréchet	B-Algorithm
spaces	I-Algorithm
to	O
be	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
A	O
TVS	O
embedding	O
or	O
a	O
topological	O
monomorphism	B-Application
is	O
an	O
injective	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
Equivalently	O
,	O
a	O
TVS-embedding	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
that	O
is	O
also	O
a	O
topological	O
embedding	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
between	O
TVSs	O
and	O
note	O
that	O
can	O
be	O
decomposed	O
into	O
the	O
composition	O
of	O
the	O
following	O
canonical	O
linear	B-Architecture
maps	I-Architecture
:	O
</s>
<s>
where	O
is	O
the	O
canonical	O
quotient	O
map	O
and	O
is	O
the	O
inclusion	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
open	O
mapping	O
theorem	O
,	O
also	O
known	O
as	O
Banach	O
's	O
homomorphism	O
theorem	O
,	O
gives	O
a	O
sufficient	O
condition	O
for	O
a	O
continuous	O
linear	B-Architecture
operator	I-Architecture
between	O
complete	B-Algorithm
metrizable	I-Algorithm
TVSs	I-Algorithm
to	O
be	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
<s>
Every	O
continuous	O
linear	O
functional	O
on	O
a	O
TVS	O
is	O
a	O
topological	B-Algorithm
homomorphism	I-Algorithm
.	O
</s>
