<s>
In	O
mathematics	O
,	O
more	O
precisely	O
in	O
functional	B-Application
analysis	I-Application
,	O
an	O
energetic	B-Algorithm
space	I-Algorithm
is	O
,	O
intuitively	O
,	O
a	O
subspace	O
of	O
a	O
given	O
real	O
Hilbert	O
space	O
equipped	O
with	O
a	O
new	O
"	O
energetic	O
"	O
inner	O
product	O
.	O
</s>
<s>
The	O
motivation	O
for	O
the	O
name	O
comes	O
from	O
physics	O
,	O
as	O
in	O
many	O
physical	O
problems	O
the	O
energy	O
of	O
a	O
system	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
energetic	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
.	O
</s>
<s>
The	O
set	O
together	O
with	O
the	O
energetic	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
is	O
a	O
pre-Hilbert	O
space	O
.	O
</s>
<s>
The	O
energetic	B-Algorithm
space	I-Algorithm
is	O
defined	O
as	O
the	O
completion	O
of	O
in	O
the	O
energetic	B-Algorithm
norm	I-Algorithm
.	O
</s>
<s>
can	O
be	O
considered	O
a	O
subset	O
of	O
the	O
original	O
Hilbert	O
space	O
since	O
any	O
Cauchy	O
sequence	O
in	O
the	O
energetic	B-Algorithm
norm	I-Algorithm
is	O
also	O
Cauchy	O
in	O
the	O
norm	O
of	O
(	O
this	O
follows	O
from	O
the	O
strong	O
monotonicity	O
property	O
of	O
)	O
.	O
</s>
<s>
where	O
and	O
are	O
sequences	O
in	O
Y	O
that	O
converge	O
to	O
points	O
in	O
in	O
the	O
energetic	B-Algorithm
norm	I-Algorithm
.	O
</s>
<s>
by	O
the	O
definition	O
of	O
the	O
energetic	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
.	O
</s>
<s>
Let	O
the	O
vertical	O
outer	O
force	O
density	O
at	O
each	O
point	O
on	O
the	O
string	O
be	O
,	O
where	O
is	O
a	O
unit	O
vector	O
pointing	O
vertically	O
and	O
Let	O
be	O
the	O
deflection	B-Algorithm
of	O
the	O
string	O
at	O
the	O
point	O
under	O
the	O
influence	O
of	O
the	O
force	O
.	O
</s>
<s>
To	O
study	O
this	O
equation	O
,	O
consider	O
the	O
space	O
that	O
is	O
,	O
the	O
Lp	O
space	O
of	O
all	O
square-integrable	B-Algorithm
functions	I-Algorithm
in	O
respect	O
to	O
the	O
Lebesgue	O
measure	O
.	O
</s>
<s>
for	O
any	O
and	O
in	O
Therefore	O
,	O
is	O
a	O
symmetric	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
so	O
it	O
is	O
half	O
of	O
the	O
energetic	B-Algorithm
inner	I-Algorithm
product	I-Algorithm
of	O
with	O
itself	O
.	O
</s>
<s>
For	O
example	O
,	O
one	O
might	O
let	O
be	O
a	O
continuous	O
piecewise	O
linear	O
function	O
in	O
the	O
energetic	B-Algorithm
space	I-Algorithm
,	O
which	O
gives	O
the	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
<s>
The	O
energetic	B-Algorithm
norm	I-Algorithm
turns	O
out	O
to	O
be	O
the	O
natural	O
norm	O
in	O
which	O
to	O
measure	O
the	O
error	O
between	O
and	O
,	O
see	O
Céa	O
's	O
lemma	O
.	O
</s>
