<s>
In	O
mathematics	O
,	O
a	O
square-integrable	B-Algorithm
function	I-Algorithm
,	O
also	O
called	O
a	O
quadratically	B-Algorithm
integrable	I-Algorithm
function	I-Algorithm
or	O
function	O
or	O
square-summable	B-Algorithm
function	I-Algorithm
,	O
is	O
a	O
real	O
-	O
or	O
complex-valued	O
measurable	O
function	O
for	O
which	O
the	O
integral	O
of	O
the	O
square	O
of	O
the	O
absolute	O
value	O
is	O
finite	O
.	O
</s>
<s>
Thus	O
,	O
square-integrability	B-Algorithm
on	O
the	O
real	O
line	O
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
One	O
may	O
also	O
speak	O
of	O
quadratic	B-Algorithm
integrability	I-Algorithm
over	O
bounded	O
intervals	O
such	O
as	O
for	O
.	O
</s>
<s>
The	O
vector	O
space	O
of	O
(	O
equivalence	O
classes	O
of	O
)	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
(	O
with	O
respect	O
to	O
Lebesgue	O
measure	O
)	O
forms	O
the	O
space	O
with	O
Among	O
the	O
spaces	O
,	O
the	O
class	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
is	O
unique	O
in	O
being	O
compatible	O
with	O
an	O
inner	O
product	O
,	O
which	O
allows	O
notions	O
like	O
angle	O
and	O
orthogonality	O
to	O
be	O
defined	O
.	O
</s>
<s>
Along	O
with	O
this	O
inner	O
product	O
,	O
the	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
form	O
a	O
Hilbert	O
space	O
,	O
since	O
all	O
of	O
the	O
spaces	O
are	O
complete	O
under	O
their	O
respective	O
-norms	O
.	O
</s>
<s>
and	O
are	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
,	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
form	O
a	O
complete	O
metric	O
space	O
under	O
the	O
metric	O
induced	O
by	O
the	O
inner	O
product	O
defined	O
above	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
space	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
is	O
a	O
Banach	O
space	O
,	O
under	O
the	O
metric	O
induced	O
by	O
the	O
norm	O
,	O
which	O
in	O
turn	O
is	O
induced	O
by	O
the	O
inner	O
product	O
.	O
</s>
<s>
Note	O
that	O
denotes	O
the	O
set	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
,	O
but	O
no	O
selection	O
of	O
metric	O
,	O
norm	O
or	O
inner	O
product	O
are	O
specified	O
by	O
this	O
notation	O
.	O
</s>
<s>
The	O
function	O
defined	O
on	O
is	O
square-integrable	B-Algorithm
.	O
</s>
<s>
Bounded	O
functions	O
,	O
defined	O
on	O
are	O
square-integrable	B-Algorithm
.	O
</s>
