<s>
In	O
mathematics	O
specifically	O
,	O
in	O
operator	O
theory	O
a	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
or	O
partially	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
is	O
a	O
type	O
of	O
partially	B-Algorithm
defined	I-Algorithm
function	O
.	O
</s>
<s>
In	O
a	O
topological	B-Architecture
sense	O
,	O
it	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
that	O
is	O
defined	O
"	O
almost	O
everywhere	O
"	O
.	O
</s>
<s>
Densely	B-Algorithm
defined	I-Algorithm
operators	I-Algorithm
often	O
arise	O
in	O
functional	B-Application
analysis	I-Application
as	O
operations	O
that	O
one	O
would	O
like	O
to	O
apply	O
to	O
a	O
larger	O
class	O
of	O
objects	O
than	O
those	O
for	O
which	O
they	O
a	O
priori	O
"	O
make	O
sense	O
"	O
.	O
</s>
<s>
The	O
Paley	O
–	O
Wiener	O
integral	O
,	O
on	O
the	O
other	O
hand	O
,	O
is	O
an	O
example	O
of	O
a	O
continuous	O
extension	O
of	O
a	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
In	O
any	O
abstract	O
Wiener	O
space	O
with	O
adjoint	O
there	O
is	O
a	O
natural	O
continuous	B-Algorithm
linear	I-Algorithm
operator	I-Algorithm
(	O
in	O
fact	O
it	O
is	O
the	O
inclusion	O
,	O
and	O
is	O
an	O
isometry	O
)	O
from	O
to	O
under	O
which	O
goes	O
to	O
the	O
equivalence	O
class	O
of	O
in	O
It	O
can	O
be	O
shown	O
that	O
is	O
dense	O
in	O
Since	O
the	O
above	O
inclusion	O
is	O
continuous	O
,	O
there	O
is	O
a	O
unique	O
continuous	O
linear	O
extension	O
of	O
the	O
inclusion	O
to	O
the	O
whole	O
of	O
This	O
extension	O
is	O
the	O
Paley	O
–	O
Wiener	O
map	O
.	O
</s>
