• Source: Energetic space

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.




Energetic space


Formally, consider a real Hilbert space



X


{\displaystyle X}

with the inner product



(
⋅

|

⋅
)


{\displaystyle (\cdot |\cdot )}

and the norm



‖
⋅
‖


{\displaystyle \|\cdot \|}

. Let



Y


{\displaystyle Y}

be a linear subspace of



X


{\displaystyle X}

and



B
:
Y
→
X


{\displaystyle B:Y\to X}

be a strongly monotone symmetric linear operator, that is, a linear operator satisfying




(
B
u

|

v
)
=
(
u

|

B
v
)



{\displaystyle (Bu|v)=(u|Bv)\,}

for all



u
,
v


{\displaystyle u,v}

in



Y


{\displaystyle Y}





(
B
u

|

u
)
≥
c
‖
u

‖

2




{\displaystyle (Bu|u)\geq c\|u\|^{2}}

for some constant



c
>
0


{\displaystyle c>0}

and all



u


{\displaystyle u}

in



Y
.


{\displaystyle Y.}


The energetic inner product is defined as




(
u

|

v

)

E


=
(
B
u

|

v
)



{\displaystyle (u|v)_{E}=(Bu|v)\,}

for all



u
,
v


{\displaystyle u,v}

in



Y


{\displaystyle Y}


and the energetic norm is




‖
u

‖

E


=
(
u

|

u

)

E



1
2






{\displaystyle \|u\|_{E}=(u|u)_{E}^{\frac {1}{2}}\,}

for all



u


{\displaystyle u}

in



Y
.


{\displaystyle Y.}


The set



Y


{\displaystyle Y}

together with the energetic inner product is a pre-Hilbert space. The energetic space




X

E




{\displaystyle X_{E}}

is defined as the completion of



Y


{\displaystyle Y}

in the energetic norm.




X

E




{\displaystyle X_{E}}

can be considered a subset of the original Hilbert space



X
,


{\displaystyle X,}

since any Cauchy sequence in the energetic norm is also Cauchy in the norm of



X


{\displaystyle X}

(this follows from the strong monotonicity property of



B


{\displaystyle B}

).
The energetic inner product is extended from



Y


{\displaystyle Y}

to




X

E




{\displaystyle X_{E}}

by




(
u

|

v

)

E


=

lim

n
→
∞


(

u

n



|


v

n



)

E




{\displaystyle (u|v)_{E}=\lim _{n\to \infty }(u_{n}|v_{n})_{E}}


where



(

u

n


)


{\displaystyle (u_{n})}

and



(

v

n


)


{\displaystyle (v_{n})}

are sequences in Y that converge to points in




X

E




{\displaystyle X_{E}}

in the energetic norm.




Energetic extension


The operator



B


{\displaystyle B}

admits an energetic extension




B

E




{\displaystyle B_{E}}







B

E


:

X

E


→

X

E


∗




{\displaystyle B_{E}:X_{E}\to X_{E}^{*}}


defined on




X

E




{\displaystyle X_{E}}

with values in the dual space




X

E


∗




{\displaystyle X_{E}^{*}}

that is given by the formula




⟨

B

E


u

|

v

⟩

E


=
(
u

|

v

)

E




{\displaystyle \langle B_{E}u|v\rangle _{E}=(u|v)_{E}}

for all



u
,
v


{\displaystyle u,v}

in




X

E


.


{\displaystyle X_{E}.}


Here,



⟨
⋅

|

⋅

⟩

E




{\displaystyle \langle \cdot |\cdot \rangle _{E}}

denotes the duality bracket between




X

E


∗




{\displaystyle X_{E}^{*}}

and




X

E


,


{\displaystyle X_{E},}

so



⟨

B

E


u

|

v

⟩

E




{\displaystyle \langle B_{E}u|v\rangle _{E}}

actually denotes



(

B

E


u
)
(
v
)
.


{\displaystyle (B_{E}u)(v).}


If



u


{\displaystyle u}

and



v


{\displaystyle v}

are elements in the original subspace



Y
,


{\displaystyle Y,}

then




⟨

B

E


u

|

v

⟩

E


=
(
u

|

v

)

E


=
(
B
u

|

v
)
=
⟨
u

|

B

|

v
⟩


{\displaystyle \langle B_{E}u|v\rangle _{E}=(u|v)_{E}=(Bu|v)=\langle u|B|v\rangle }


by the definition of the energetic inner product. If one views



B
u
,


{\displaystyle Bu,}

which is an element in



X
,


{\displaystyle X,}

as an element in the dual




X

∗




{\displaystyle X^{*}}

via the Riesz representation theorem, then



B
u


{\displaystyle Bu}

will also be in the dual




X

E


∗




{\displaystyle X_{E}^{*}}

(by the strong monotonicity property of



B


{\displaystyle B}

). Via these identifications, it follows from the above formula that




B

E


u
=
B
u
.


{\displaystyle B_{E}u=Bu.}

In different words, the original operator



B
:
Y
→
X


{\displaystyle B:Y\to X}

can be viewed as an operator



B
:
Y
→

X

E


∗


,


{\displaystyle B:Y\to X_{E}^{*},}

and then




B

E


:

X

E


→

X

E


∗




{\displaystyle B_{E}:X_{E}\to X_{E}^{*}}

is simply the function extension of



B


{\displaystyle B}

from



Y


{\displaystyle Y}

to




X

E


.


{\displaystyle X_{E}.}






An example from physics



Consider a string whose endpoints are fixed at two points



a
<
b


{\displaystyle a
on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point



x


{\displaystyle x}





(
a
≤
x
≤
b
)


{\displaystyle (a\leq x\leq b)}

on the string be



f
(
x
)

e



{\displaystyle f(x)\mathbf {e} }

, where




e



{\displaystyle \mathbf {e} }

is a unit vector pointing vertically and



f
:
[
a
,
b
]
→

R

.


{\displaystyle f:[a,b]\to \mathbb {R} .}

Let



u
(
x
)


{\displaystyle u(x)}

be the deflection of the string at the point



x


{\displaystyle 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


{\displaystyle {\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx}


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
.


{\displaystyle F(u)={\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx-\int _{a}^{b}\!u(x)f(x)\,dx.}


The deflection



u
(
x
)


{\displaystyle u(x)}

minimizing the potential energy will satisfy the differential equation




−

u
″

=
f



{\displaystyle -u''=f\,}


with boundary conditions




u
(
a
)
=
u
(
b
)
=
0.



{\displaystyle u(a)=u(b)=0.\,}


To study this equation, consider the space



X
=

L

2


(
a
,
b
)
,


{\displaystyle X=L^{2}(a,b),}

that is, the Lp space of all square-integrable functions



u
:
[
a
,
b
]
→

R



{\displaystyle u:[a,b]\to \mathbb {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
,


{\displaystyle (u|v)=\int _{a}^{b}\!u(x)v(x)\,dx,}


with the norm being given by




‖
u
‖
=


(
u

|

u
)


.


{\displaystyle \|u\|={\sqrt {(u|u)}}.}


Let



Y


{\displaystyle Y}

be the set of all twice continuously differentiable functions



u
:
[
a
,
b
]
→

R



{\displaystyle u:[a,b]\to \mathbb {R} }

with the boundary conditions



u
(
a
)
=
u
(
b
)
=
0.


{\displaystyle u(a)=u(b)=0.}

Then



Y


{\displaystyle Y}

is a linear subspace of



X
.


{\displaystyle X.}


Consider the operator



B
:
Y
→
X


{\displaystyle B:Y\to X}

given by the formula




B
u
=
−

u
″

,



{\displaystyle Bu=-u'',\,}


so the deflection satisfies the equation



B
u
=
f
.


{\displaystyle Bu=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
)


{\displaystyle (Bu|v)=-\int _{a}^{b}\!u''(x)v(x)\,dx=\int _{a}^{b}u'(x)v'(x)=(u|Bv)}


for any



u


{\displaystyle u}

and



v


{\displaystyle v}

in



Y
.


{\displaystyle Y.}

Therefore,



B


{\displaystyle B}

is a symmetric linear operator.




B


{\displaystyle B}

is also strongly monotone, since, by the Friedrichs's inequality




‖
u

‖

2


=

∫

a


b



u

2


(
x
)

d
x
≤
C

∫

a


b



u
′

(
x

)

2



d
x
=
C

(
B
u

|

u
)


{\displaystyle \|u\|^{2}=\int _{a}^{b}u^{2}(x)\,dx\leq C\int _{a}^{b}u'(x)^{2}\,dx=C\,(Bu|u)}


for some



C
>
0.


{\displaystyle C>0.}


The energetic space in respect to the operator



B


{\displaystyle B}

is then the Sobolev space




H

0


1


(
a
,
b
)
.


{\displaystyle 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


,


{\displaystyle {\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx={\frac {1}{2}}(u|u)_{E},}


so it is half of the energetic inner product of



u


{\displaystyle u}

with itself.
To calculate the deflection



u


{\displaystyle u}

minimizing the total potential energy



F
(
u
)


{\displaystyle F(u)}

of the string, one writes this problem in the form




(
u

|

v

)

E


=
(
f

|

v
)



{\displaystyle (u|v)_{E}=(f|v)\,}

for all



v


{\displaystyle v}

in




X

E




{\displaystyle X_{E}}

.
Next, one usually approximates



u


{\displaystyle u}

by some




u

h




{\displaystyle u_{h}}

, a function in a finite-dimensional subspace of the true solution space. For example, one might let




u

h




{\displaystyle u_{h}}

be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation




u

h




{\displaystyle u_{h}}

can be computed by solving a system of linear equations.
The energetic norm turns out to be the natural norm in which to measure the error between



u


{\displaystyle u}

and




u

h




{\displaystyle u_{h}}

, see Céa's lemma.




See also


Inner product space
Positive-definite kernel




References


Zeidler, Eberhard (1995). Applied functional analysis: applications to mathematical physics. New York: Springer-Verlag. ISBN 0-387-94442-7.
Johnson, Claes (1987). Numerical solution of partial differential equations by the finite element method. Cambridge University Press. ISBN 0-521-34514-6.

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