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 $\Vert \cdot \Vert$. 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 $Y$
• $(Bu|u)\ge c\Vert u{\Vert }^2$ 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 normTemplate:Anchor is

$\Vert u{\Vert }_E=(u|u{)}_E^{\frac{1}{2}}$ 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

$(u|v{)}_E=\underset{n\to \mathrm{\infty }}{\lim }(u_n|v_n{)}_E$

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 $B_E$

$B_E:X_E\to X_E^{\ast }$

defined on $X_E$ with values in the dual space $X_E^{\ast }$ that is given by the formula

$⟨B_{E}u|v{⟩}_E=(u|v{)}_E$ for all $u,v$ in $X_E.$

Here, $⟨\cdot |\cdot {⟩}_E$ denotes the duality bracket between $X_E^{\ast }$ 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

$⟨B_{E}u|v{⟩}_E=(u|v{)}_E=(Bu|v)=⟨u|B|v⟩$

by the definition of the energetic inner product. If one views $Bu,$ which is an element in $X,$ as an element in the dual $X^{\ast }$ via the Riesz representation theorem, then $Bu$ will also be in the dual $X_E^{\ast }$ (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 $B:Y\to X_E^{\ast },$ and then $B_E:X_E\to X_E^{\ast }$ is simply the function extension of $B$ from $Y$ to $X_E.$

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 $a<b$ on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point $x$ $(a\le x\le b)$ on the string be $f(x)𝐞$, where $𝐞$ is a unit vector pointing vertically and $f:[a,b]\to ℝ.$ 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

$\frac{1}{2}{\int }_a^b{u}^{\prime }(x{)}^{2}dx$

and the total potential energy of the string is

$F(u)=\frac{1}{2}{\int }_a^b{u}^{\prime }(x{)}^{2}dx-{\int }_a^{b}u(x)f(x)dx.$

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 $u:[a,b]\to ℝ$ 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

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

Let $Y$ be the set of all twice continuously differentiable functions $u:[a,b]\to ℝ$ 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

$(Bu|v)=-{\int }_a^b{u}^{″}(x)v(x)dx={\int }_a^b{u}^{\prime }(x)v^{\prime }(x)=(u|Bv)$

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

$\Vert u{\Vert }^2={\int }_a^b{u}^2(x)dx\le C{\int }_a^b{u}^{\prime }(x{)}^{2}dx=C(Bu|u)$

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

$\frac{1}{2}{\int }_a^b{u}^{\prime }(x{)}^{2}dx=\frac{1}{2}(u|u{)}_E,$

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.

See also

• Inner product space
• Positive-definite kernel

References

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