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 ${\displaystyle X}$  with the inner product ${\displaystyle (\cdot |\cdot )}$  and the norm ${\displaystyle \|\cdot \|}$ . Let ${\displaystyle Y}$  be a linear subspace of ${\displaystyle X}$  and ${\displaystyle B:Y\to X}$  be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

• ${\displaystyle (Bu|v)=(u|Bv)\,}$  for all ${\displaystyle u,v}$  in ${\displaystyle Y}$ 
• ${\displaystyle (Bu|u)\geq c\|u\|^{2}}$  for some constant ${\displaystyle c>0}$  and all ${\displaystyle u}$  in ${\displaystyle Y.}$ 

The energetic inner product is defined as

${\displaystyle (u|v)_{E}=(Bu|v)\,}$  for all ${\displaystyle u,v}$  in ${\displaystyle Y}$ 

and the energetic norm is

${\displaystyle \|u\|_{E}=(u|u)_{E}^{\frac {1}{2}}\,}$  for all ${\displaystyle u}$  in ${\displaystyle Y.}$ 

The set ${\displaystyle Y}$  together with the energetic inner product is a pre-Hilbert space. The energetic space ${\displaystyle X_{E}}$  is defined as the completion of ${\displaystyle Y}$  in the energetic norm. ${\displaystyle X_{E}}$  can be considered a subset of the original Hilbert space ${\displaystyle X,}$  since any Cauchy sequence in the energetic norm is also Cauchy in the norm of ${\displaystyle X}$  (this follows from the strong monotonicity property of ${\displaystyle B}$ ).

The energetic inner product is extended from ${\displaystyle Y}$  to ${\displaystyle X_{E}}$  by

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

where ${\displaystyle (u_{n})}$  and ${\displaystyle (v_{n})}$  are sequences in Y that converge to points in ${\displaystyle X_{E}}$  in the energetic norm.

Energetic extension

The operator ${\displaystyle B}$  admits an energetic extension ${\displaystyle B_{E}}$ 

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

defined on ${\displaystyle X_{E}}$  with values in the dual space ${\displaystyle X_{E}^{*}}$  that is given by the formula

${\displaystyle \langle B_{E}u|v\rangle _{E}=(u|v)_{E}}$  for all ${\displaystyle u,v}$  in ${\displaystyle X_{E}.}$ 

Here, ${\displaystyle \langle \cdot |\cdot \rangle _{E}}$  denotes the duality bracket between ${\displaystyle X_{E}^{*}}$  and ${\displaystyle X_{E},}$  so ${\displaystyle \langle B_{E}u|v\rangle _{E}}$  actually denotes ${\displaystyle (B_{E}u)(v).}$ 

If ${\displaystyle u}$  and ${\displaystyle v}$  are elements in the original subspace ${\displaystyle Y,}$  then

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

by the definition of the energetic inner product. If one views ${\displaystyle Bu,}$  which is an element in ${\displaystyle X,}$  as an element in the dual ${\displaystyle X^{*}}$  via the Riesz representation theorem, then ${\displaystyle Bu}$  will also be in the dual ${\displaystyle X_{E}^{*}}$  (by the strong monotonicity property of ${\displaystyle B}$ ). Via these identifications, it follows from the above formula that ${\displaystyle B_{E}u=Bu.}$  In different words, the original operator ${\displaystyle B:Y\to X}$  can be viewed as an operator ${\displaystyle B:Y\to X_{E}^{*},}$  and then ${\displaystyle B_{E}:X_{E}\to X_{E}^{*}}$  is simply the function extension of ${\displaystyle B}$  from ${\displaystyle Y}$  to ${\displaystyle X_{E}.}$ 

An example from physics

 

A string with fixed endpoints under the influence of a force pointing down.

Consider a string whose endpoints are fixed at two points ${\displaystyle a<b}$  on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point ${\displaystyle x}$  ${\displaystyle (a\leq x\leq b)}$  on the string be ${\displaystyle f(x)\mathbf {e} }$ , where ${\displaystyle \mathbf {e} }$  is a unit vector pointing vertically and ${\displaystyle f:[a,b]\to \mathbb {R} .}$  Let ${\displaystyle u(x)}$  be the deflection of the string at the point ${\displaystyle x}$  under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is

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

and the total potential energy of the string is

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

The deflection ${\displaystyle u(x)}$  minimizing the potential energy will satisfy the differential equation

${\displaystyle -u''=f\,}$ 

with boundary conditions

${\displaystyle u(a)=u(b)=0.\,}$ 

To study this equation, consider the space ${\displaystyle X=L^{2}(a,b),}$  that is, the Lp space of all square-integrable functions ${\displaystyle u:[a,b]\to \mathbb {R} }$  in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

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

with the norm being given by

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

Let ${\displaystyle Y}$  be the set of all twice continuously differentiable functions ${\displaystyle u:[a,b]\to \mathbb {R} }$  with the boundary conditions ${\displaystyle u(a)=u(b)=0.}$  Then ${\displaystyle Y}$  is a linear subspace of ${\displaystyle X.}$ 

Consider the operator ${\displaystyle B:Y\to X}$  given by the formula

${\displaystyle Bu=-u'',\,}$ 

so the deflection satisfies the equation ${\displaystyle Bu=f.}$  Using integration by parts and the boundary conditions, one can see that

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

for any ${\displaystyle u}$  and ${\displaystyle v}$  in ${\displaystyle Y.}$  Therefore, ${\displaystyle B}$  is a symmetric linear operator.

${\displaystyle B}$  is also strongly monotone, since, by the Friedrichs's inequality

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

for some ${\displaystyle C>0.}$ 

The energetic space in respect to the operator ${\displaystyle B}$  is then the Sobolev space ${\displaystyle H_{0}^{1}(a,b).}$  We see that the elastic energy of the string which motivated this study is

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

so it is half of the energetic inner product of ${\displaystyle u}$  with itself.

To calculate the deflection ${\displaystyle u}$  minimizing the total potential energy ${\displaystyle F(u)}$  of the string, one writes this problem in the form

${\displaystyle (u|v)_{E}=(f|v)\,}$  for all ${\displaystyle v}$  in ${\displaystyle X_{E}}$ .

Next, one usually approximates ${\displaystyle u}$  by some ${\displaystyle u_{h}}$ , a function in a finite-dimensional subspace of the true solution space. For example, one might let ${\displaystyle u_{h}}$  be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation ${\displaystyle u_{h}}$  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 ${\displaystyle u}$  and ${\displaystyle u_{h}}$ , 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.