<s>
Weak	B-Algorithm
formulations	I-Algorithm
are	O
important	O
tools	O
for	O
the	O
analysis	O
of	O
mathematical	O
equations	O
that	O
permit	O
the	O
transfer	O
of	O
concepts	O
of	O
linear	B-Language
algebra	I-Language
to	O
solve	O
problems	O
in	O
other	O
fields	O
such	O
as	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
In	O
a	O
weak	B-Algorithm
formulation	I-Algorithm
,	O
equations	O
or	O
conditions	O
are	O
no	O
longer	O
required	O
to	O
hold	O
absolutely	O
(	O
and	O
this	O
is	O
not	O
even	O
well	O
defined	O
)	O
and	O
has	O
instead	O
weak	O
solutions	O
only	O
with	O
respect	O
to	O
certain	O
"	O
test	O
vectors	O
"	O
or	O
"	O
test	O
functions	O
"	O
.	O
</s>
<s>
The	O
Lax	B-Algorithm
–	I-Algorithm
Milgram	I-Algorithm
theorem	I-Algorithm
,	O
named	O
after	O
Peter	O
Lax	O
and	O
Arthur	O
Milgram	O
who	O
proved	O
it	O
in	O
1954	O
,	O
provides	O
weak	B-Algorithm
formulations	I-Algorithm
for	O
certain	O
systems	O
on	O
Hilbert	O
spaces	O
.	O
</s>
<s>
Now	O
,	O
let	O
and	O
be	O
a	O
linear	B-Architecture
mapping	I-Architecture
.	O
</s>
<s>
to	O
derive	O
the	O
weak	B-Algorithm
formulation	I-Algorithm
.	O
</s>
<s>
This	O
is	O
what	O
is	O
usually	O
called	O
the	O
weak	B-Algorithm
formulation	I-Algorithm
of	O
Poisson	O
's	O
equation	O
.	O
</s>
<s>
Functions	O
in	O
the	O
solution	O
space	O
must	O
be	O
zero	O
on	O
the	O
boundary	O
,	O
and	O
have	O
square-integrable	O
derivatives	B-Algorithm
.	O
</s>
<s>
This	O
is	O
a	O
formulation	O
of	O
the	O
Lax	B-Algorithm
–	I-Algorithm
Milgram	I-Algorithm
theorem	I-Algorithm
which	O
relies	O
on	O
properties	O
of	O
the	O
symmetric	O
part	O
of	O
the	O
bilinear	O
form	O
.	O
</s>
<s>
Here	O
,	O
application	O
of	O
the	O
Lax	B-Algorithm
–	I-Algorithm
Milgram	I-Algorithm
theorem	I-Algorithm
is	O
a	O
stronger	O
result	O
than	O
is	O
needed	O
.	O
</s>
