<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
the	O
Peano	B-Algorithm
kernel	I-Algorithm
theorem	I-Algorithm
is	O
a	O
general	O
result	O
on	O
error	O
bounds	O
for	O
a	O
wide	O
class	O
of	O
numerical	B-General_Concept
approximations	I-General_Concept
(	O
such	O
as	O
numerical	B-Algorithm
quadratures	I-Algorithm
)	O
,	O
defined	O
in	O
terms	O
of	O
linear	B-Algorithm
functionals	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
the	O
space	O
of	O
all	O
functions	O
that	O
are	O
differentiable	O
on	O
that	O
are	O
of	O
bounded	O
variation	O
on	O
,	O
and	O
let	O
be	O
a	O
linear	B-Algorithm
functional	I-Algorithm
on	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
main	O
application	O
of	O
the	O
Peano	B-Algorithm
kernel	I-Algorithm
theorem	I-Algorithm
is	O
to	O
bound	O
the	O
error	O
of	O
an	O
approximation	O
that	O
is	O
exact	O
for	O
all	O
.	O
</s>
<s>
defining	O
as	O
the	O
error	O
of	O
the	O
approximation	O
,	O
using	O
the	O
linearity	B-Architecture
of	O
together	O
with	O
exactness	O
for	O
to	O
annihilate	O
all	O
but	O
the	O
final	O
term	O
on	O
the	O
right-hand	O
side	O
,	O
and	O
using	O
the	O
notation	O
to	O
remove	O
the	O
-dependence	O
from	O
the	O
integral	O
limits	O
.	O
</s>
