<s>
In	O
functional	B-Application
analysis	I-Application
,	O
it	O
is	O
often	O
convenient	O
to	O
define	O
a	O
linear	B-Architecture
transformation	I-Architecture
on	O
a	O
complete	O
,	O
normed	O
vector	O
space	O
by	O
first	O
defining	O
a	O
linear	B-Architecture
transformation	I-Architecture
on	O
a	O
dense	O
subset	O
of	O
and	O
then	O
continuously	O
extending	O
to	O
the	O
whole	O
space	O
via	O
the	O
theorem	O
below	O
.	O
</s>
<s>
The	O
resulting	O
extension	O
remains	O
linear	B-Architecture
and	O
bounded	O
,	O
and	O
is	O
thus	O
continuous	O
,	O
which	O
makes	O
it	O
a	O
continuous	B-Algorithm
linear	I-Algorithm
extension	I-Algorithm
.	O
</s>
<s>
This	O
procedure	O
is	O
known	O
as	O
continuous	B-Algorithm
linear	I-Algorithm
extension	I-Algorithm
.	O
</s>
<s>
Let	O
denote	O
the	O
space	O
of	O
bounded	O
,	O
piecewise	B-Algorithm
continuous	I-Algorithm
functions	O
on	O
that	O
are	O
continuous	O
from	O
the	O
right	O
,	O
along	O
with	O
the	O
norm	O
.	O
</s>
<s>
The	O
above	O
theorem	O
can	O
be	O
used	O
to	O
extend	O
a	O
bounded	O
linear	B-Architecture
transformation	I-Architecture
to	O
a	O
bounded	O
linear	B-Architecture
transformation	I-Architecture
from	O
to	O
if	O
is	O
dense	O
in	O
If	O
is	O
not	O
dense	O
in	O
then	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
may	O
sometimes	O
be	O
used	O
to	O
show	O
that	O
an	O
extension	O
exists	O
.	O
</s>
