In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.
Formally, consider a real Hilbert space
X
with the inner product
(
⋅
|
⋅
)
and the norm
∥
⋅
∥
. Let
Y
be a linear subspace of
X
and
B
:
Y
→
X
be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
(
B
u
|
v
)
=
(
u
|
B
v
)
for all
u
,
v
in
Y
(
B
u
|
u
)
≥
c
∥
u
∥
2
for some constant
c
>
0
and all
u
in
Y
.
The energetic inner product is defined as
(
u
|
v
)
E
=
(
B
u
|
v
)
for all
u
,
v
in
Y
and the energetic norm is
∥
u
∥
E
=
(
u
|
u
)
E
1
2
for all
u
in
Y
.
The set
Y
together with the energetic inner product is a pre-Hilbert space. The energetic space
X
E
is defined as the completion of
Y
in the energetic norm.
X
E
can be considered a subset of the original Hilbert space
X
,
since any Cauchy sequence in the energetic norm is also Cauchy in the norm of
X
(this follows from the strong monotonicity property of
B
).
The energetic inner product is extended from
Y
to
X
E
by
(
u
|
v
)
E
=
lim
n
→
∞
(
u
n
|
v
n
)
E
where
(
u
n
)
and
(
v
n
)
are sequences in Y that converge to points in
X
E
in the energetic norm.
The operator
B
admits an energetic extension
B
E
B
E
:
X
E
→
X
E
∗
defined on
X
E
with values in the dual space
X
E
∗
that is given by the formula
⟨
B
E
u
|
v
⟩
E
=
(
u
|
v
)
E
for all
u
,
v
in
X
E
.
Here,
⟨
⋅
|
⋅
⟩
E
denotes the duality bracket between
X
E
∗
and
X
E
,
so
⟨
B
E
u
|
v
⟩
E
actually denotes
(
B
E
u
)
(
v
)
.
If
u
and
v
are elements in the original subspace
Y
,
then
⟨
B
E
u
|
v
⟩
E
=
(
u
|
v
)
E
=
(
B
u
|
v
)
=
⟨
u
|
B
|
v
⟩
by the definition of the energetic inner product. If one views
B
u
,
which is an element in
X
,
as an element in the dual
X
∗
via the Riesz representation theorem, then
B
u
will also be in the dual
X
E
∗
(by the strong monotonicity property of
B
). Via these identifications, it follows from the above formula that
B
E
u
=
B
u
.
In different words, the original operator
B
:
Y
→
X
can be viewed as an operator
B
:
Y
→
X
E
∗
,
and then
B
E
:
X
E
→
X
E
∗
is simply the function extension of
B
from
Y
to
X
E
.
Consider a string whose endpoints are fixed at two points
a
<
b
on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point
x
(
a
≤
x
≤
b
)
on the string be
f
(
x
)
e
, where
e
is a unit vector pointing vertically and
f
:
[
a
,
b
]
→
R
.
Let
u
(
x
)
be the deflection of the string at the point
x
under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is
1
2
∫
a
b
u
′
(
x
)
2
d
x
and the total potential energy of the string is
F
(
u
)
=
1
2
∫
a
b
u
′
(
x
)
2
d
x
−
∫
a
b
u
(
x
)
f
(
x
)
d
x
.
The deflection
u
(
x
)
minimizing the potential energy will satisfy the differential equation
−
u
″
=
f
with boundary conditions
u
(
a
)
=
u
(
b
)
=
0.
To study this equation, consider the space
X
=
L
2
(
a
,
b
)
,
that is, the Lp space of all square integrable functions
u
:
[
a
,
b
]
→
R
in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
(
u
|
v
)
=
∫
a
b
u
(
x
)
v
(
x
)
d
x
,
with the norm being given by
∥
u
∥
=
(
u
|
u
)
.
Let
Y
be the set of all twice continuously differentiable functions
u
:
[
a
,
b
]
→
R
with the boundary conditions
u
(
a
)
=
u
(
b
)
=
0.
Then
Y
is a linear subspace of
X
.
Consider the operator
B
:
Y
→
X
given by the formula
B
u
=
−
u
″
,
so the deflection satisfies the equation
B
u
=
f
.
Using integration by parts and the boundary conditions, one can see that
(
B
u
|
v
)
=
−
∫
a
b
u
″
(
x
)
v
(
x
)
d
x
=
∫
a
b
u
′
(
x
)
v
′
(
x
)
=
(
u
|
B
v
)
for any
u
and
v
in
Y
.
Therefore,
B
is a symmetric linear operator.
B
is also strongly monotone, since, by the Friedrichs' inequality
∥
u
∥
2
=
∫
a
b
u
2
(
x
)
d
x
≤
C
∫
a
b
u
′
(
x
)
2
d
x
=
C
(
B
u
|
u
)
for some
C
>
0.
The energetic space in respect to the operator
B
is then the Sobolev space
H
0
1
(
a
,
b
)
.
We see that the elastic energy of the string which motivated this study is
1
2
∫
a
b
u
′
(
x
)
2
d
x
=
1
2
(
u
|
u
)
E
,
so it is half of the energetic inner product of
u
with itself.
To calculate the deflection
u
minimizing the total potential energy
F
(
u
)
of the string, one writes this problem in the form
(
u
|
v
)
E
=
(
f
|
v
)
for all
v
in
X
E
.
Next, one usually approximates
u
by some
u
h
, a function in a finite-dimensional subspace of the true solution space. For example, one might let
u
h
be a continuous piecewise-linear function in the energetic space, which gives the finite element method. The approximation
u
h
can be computed by solving a linear system of equations.
The energetic norm turns out to be the natural norm in which to measure the error between
u
and
u
h
, see Céa's lemma.
