# 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 $X$ with the inner product $(\cdot |\cdot )$ and the norm $\|\cdot \|$. Let $Y$ be a linear subspace of $X$ and $B:Y\to X$ be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
* $(Bu|v)=(u|Bv)\,$ for all $u,v$ in $Y$
* $(Bu|u)\geq c\|u\|^{2}$ for some constant $c>0$ and all $u$ in $Y.$
The energetic inner product is defined as
    $(u|v)_{E}=(Bu|v)\,$ for all $u,v$ in $Y$
and the energetic norm is
    $\|u\|_{E}=(u|u)_{E}^{\frac {1}{2}}\,$ 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
    $(u|v)_{E}=\lim _{n\to \infty }(u_{n}|v_{n})_{E}$
where $(u_{n})$ and $(v_{n})$ are sequences in Y that converge to points in $X_{E}$ in the energetic norm.

## Energetic extension

The operator $B$ admits an energetic extension $B_{E}$
    $B_{E}:X_{E}\to X_{E}^{*}$
defined on $X_{E}$ with values in the dual space $X_{E}^{*}$ that is given by the formula
    $\langle B_{E}u|v\rangle _{E}=(u|v)_{E}$ for all $u,v$ in $X_{E}.$
Here, $\langle \cdot |\cdot \rangle _{E}$ denotes the duality bracket between $X_{E}^{*}$ and $X_{E},$ so $\langle B_{E}u|v\rangle _{E}$ actually denotes $(B_{E}u)(v).$
If $u$ and $v$ are elements in the original subspace $Y,$ then
    $\langle B_{E}u|v\rangle _{E}=(u|v)_{E}=(Bu|v)=\langle u|B|v\rangle$
by the definition of the energetic inner product. If one views $Bu,$ which is an element in $X,$ as an element in the dual $X^{*}$ via the Riesz representation theorem, then $Bu$ 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 $B_{E}u=Bu.$ In different words, the original operator $B:Y\to X$ can be viewed as an operator $B:Y\to X_{E}^{*},$ and then $B_{E}:X_{E}\to X_{E}^{*}$ is simply the function extension of $B$ from $Y$ to $X_{E}.$

## An example from physics

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 $x$ $(a\leq x\leq b)$ on the string be $f(x)\mathbf {e}$, where $\mathbf {e}$ is a unit vector pointing vertically and $f:[a,b]\to \mathbb {R} .$ 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
    ${\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx$
and the total potential energy of the string is
    $F(u)={\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx-\int _{a}^{b}\!u(x)f(x)\,dx.$
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 $u:[a,b]\to \mathbb {R}$ in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
    $(u|v)=\int _{a}^{b}\!u(x)v(x)\,dx,$
with the norm being given by
    $\|u\|={\sqrt {(u|u)}}.$
Let $Y$ be the set of all twice continuously differentiable functions $u:[a,b]\to \mathbb {R}$ with the boundary conditions $u(a)=u(b)=0.$ Then $Y$ is a linear subspace of $X.$
Consider the operator $B:Y\to X$ given by the formula
    $Bu=-u'',\,$
so the deflection satisfies the equation $Bu=f.$ Using integration by parts and the boundary conditions, one can see that
    $(Bu|v)=-\int _{a}^{b}\!u''(x)v(x)\,dx=\int _{a}^{b}u'(x)v'(x)=(u|Bv)$
for any $u$ and $v$ in $Y.$ Therefore, $B$ is a symmetric linear operator.
$B$ is also strongly monotone, since, by the Friedrichs's inequality
    $\|u\|^{2}=\int _{a}^{b}u^{2}(x)\,dx\leq C\int _{a}^{b}u'(x)^{2}\,dx=C\,(Bu|u)$
for some $C>0.$
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
    ${\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx={\frac {1}{2}}(u|u)_{E},$
so it is half of the energetic inner product of $u$ 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 system of linear 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.