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In [[mathematics]], more precisely in [[functional analysis]], an '''energetic space''' is, intuitively, a subspace of a given [[real number|real]] [[Hilbert space]] equipped with a new "energetic" [[Inner product space|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 <math>X</math> with the [[Inner product space|inner product]] <math>(\cdot|\cdot)</math> and the [[norm (mathematics)|norm]] <math>\|\cdot\|</math>. Let <math>Y</math> be a linear subspace of <math>X</math> and <math>B:Y\to X</math> be a [[strongly monotone]] [[symmetric operator|symmetric]] [[linear operator]], that is, a linear operator satisfying 

* <math>(Bu|v)=(u|Bv)\, </math> for all <math>u, v</math> in <math>Y</math>
* <math>(Bu|u) \ge c\|u\|^2</math> for some constant <math>c>0</math> and all <math>u</math> in <math>Y.</math>

The '''energetic inner product''' is defined as 
:<math>(u|v)_E =(Bu|v)\,</math> for all <math>u,v</math> in <math>Y</math>
and the '''energetic norm'''{{anchor|energetic norm}} is
:<math>\|u\|_E=(u|u)^\frac{1}{2}_E \, </math> for all <math>u</math> in <math>Y.</math>

The set <math>Y</math> together with the energetic inner product is a [[pre-Hilbert space]]. The '''energetic space''' <math>X_E</math> is defined as the [[complete metric space|completion]] of <math>Y</math> in the energetic norm. <math>X_E</math> can be considered a subset of the original Hilbert space <math>X,</math> since any [[Cauchy sequence]] in the energetic norm is also Cauchy in the norm of <math>X</math> (this follows from the strong monotonicity property of <math>B</math>). 

The energetic inner product is extended from <math>Y</math> to <math>X_E</math> by
: <math> (u|v)_E = \lim_{n\to\infty} (u_n|v_n)_E</math>
where <math>(u_n)</math> and <math>(v_n)</math> are sequences in ''Y'' that converge to points in <math>X_E</math> in the energetic norm.

==Energetic extension==
The operator <math>B</math> admits an '''energetic extension''' <math>B_E</math> 

:<math>B_E:X_E\to X^*_E</math>

defined on <math>X_E</math> with values in the [[dual space]] <math>X^*_E</math> that is given by the formula 

:<math>\langle B_E u | v \rangle_E = (u|v)_E</math> for all <math>u,v</math> in <math>X_E.</math>

Here, <math>\langle \cdot |\cdot \rangle_E</math> denotes the duality bracket between <math>X^*_E</math> and <math>X_E,</math> so <math>\langle B_E u | v \rangle_E</math> actually denotes  <math>(B_E u)(v).</math>

If <math>u</math> and <math>v</math> are elements in the original subspace <math>Y,</math> then

:<math>\langle B_E u | v \rangle_E = (u|v)_E = (Bu|v) = \langle u|B|v\rangle</math>

by the definition of the energetic inner product.  If one views <math>Bu,</math> which is an element in <math>X,</math> as an element in the dual <math>X*</math> via the [[Riesz representation theorem]], then <math>Bu</math> will also be in the dual <math>X_E^*</math> (by the strong monotonicity property of <math>B</math>). Via these identifications, it follows from the above formula that <math>B_E u= Bu.</math> In different words, the original operator <math>B:Y\to X</math> can be viewed as an operator <math>B:Y\to X_E^*,</math> and then <math>B_E:X_E\to X^*_E</math> is simply the function extension of <math>B</math> from <math>Y</math> to <math>X_E.</math> <!--- 

I commented out the below text, since it is not clear what norm one uses to talk about convergence and boundedness. I will think more about it. 

That is, <math>B_E</math> is that [[linear functional]] which acts like ''B'' but has a domain of <math>X_E</math>—that is, its domain includes all limit points, ''u'', of the domain of ''B'' for which ''Bu<sub>n</sub>'' is bounded as <math>u_n\to u</math>.

--->

==An example from physics==
[[File:String illust.svg|right|thumb|A string with fixed endpoints under the influence of a force pointing down.]]
Consider a [[rope|string]]<!-- a piece of wire, so the link to [[rope|string]] is not ambiguous--> whose endpoints are fixed at two points <math>a<b</math> on the real line   (here viewed as a horizontal line). Let the vertical outer [[force density]] at each point <math>x</math> <math>(a\le x \le b)</math> on the string be <math>f(x)\mathbf{e}</math>, where  <math>\mathbf{e}</math> is a [[unit vector]] pointing vertically and <math>f:[a, b]\to \mathbb R.</math> Let <math>u(x)</math> be the [[Deflection (engineering)|deflection]] of the string at the point <math>x</math> under the influence of the force. Assuming that the deflection is small, the [[elastic energy]] of the string is 

: <math>\frac{1}{2} \int_a^b\! u'(x)^2\, dx</math>

and the total [[potential energy]] of the string is

: <math>F(u) = \frac{1}{2} \int_a^b\! u'(x)^2\,dx - \int_a^b\! u(x)f(x)\,dx.</math>

The deflection <math>u(x)</math> minimizing the potential energy will satisfy the [[differential equation]]

: <math>-u''=f\,</math>

with [[boundary conditions]]

:<math>u(a)=u(b)=0.\,</math>

To study this equation, consider the space <math>X=L^2(a, b), </math> that is, the [[Lp space]] of all [[square integrable function]]s <math>u:[a, b]\to \mathbb R</math> in respect to the [[Lebesgue measure]]. This space is Hilbert in respect to the inner product

: <math>(u|v)=\int_a^b\! u(x)v(x)\,dx,</math>

with the norm being given by 

: <math>\|u\|=\sqrt{(u|u)}.</math>

Let <math>Y</math> be the set of all [[smooth function|twice continuously differentiable functions]] <math>u:[a, b]\to \mathbb R</math> with the [[boundary conditions]]s <math>u(a)=u(b)=0.</math> Then <math>Y</math> is a linear subspace of <math>X.</math>

Consider the operator <math>B:Y\to X</math> given by the formula

: <math>Bu = -u'',\,</math>

so the deflection satisfies the equation <math>Bu=f.</math> Using  [[integration by parts]] and the boundary conditions, one can see that 

: <math>(Bu|v)=-\int_a^b\! u''(x)v(x)\, dx=\int_a^b u'(x)v'(x) = (u|Bv) </math>

for any <math>u</math> and <math>v</math> in <math>Y.</math> Therefore, <math>B</math> is a symmetric linear operator. 

<math>B</math> is also strongly monotone, since, by the [[Friedrichs' inequality]] 

: <math>\|u\|^2 = \int_a^b u^2(x)\, dx \le C \int_a^b u'(x)^2\, dx = C\,(Bu|u)</math>

for some <math>C>0.</math>

The energetic space in respect to the operator <math>B</math> is then the [[Sobolev space]] <math>H^1_0(a, b).</math> We see that the elastic energy of the string which motivated this study is 

: <math>\frac{1}{2} \int_a^b\! u'(x)^2\, dx = \frac{1}{2} (u|u)_E,</math>

so it is half of the energetic inner product of <math>u</math> with itself. 

To calculate the deflection <math>u</math> minimizing the total potential energy <math>F(u)</math> of the string, one writes this problem in the form 

:<math>(u|v)_E=(f|v)\,</math> for all <math>v</math> in <math>X_E</math>.

Next, one usually approximates <math>u</math> by some <math>u_h</math>, a function in a finite-dimensional subspace of the true solution space. For example, one might let <math>u_h</math> be a continuous [[piecewise-linear function]] in the energetic space, which gives the [[finite element method]]. The approximation <math>u_h</math> 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  <math>u</math> and <math>u_h</math>, see [[Céa's lemma]].

==See also==
* [[Inner product space]]
* [[Positive definite kernel]]

==References==
*{{cite book
 | last       = Zeidler
 | first      = Eberhard
 | title      = Applied functional analysis: applications to mathematical physics
 | publisher  = New York: Springer-Verlag
 | date       = 1995
 | pages      = 
 | isbn       = 0-387-94442-7
}}

*{{cite book
 | last       = Johnson
 | first      = Claes
 | title      = Numerical solution of partial differential equations by the finite element method
 | publisher  = Cambridge University Press
 | date       = 1987
 | pages      = 
 | isbn       = 0-521-34514-6
}}

[[Category:Functional analysis]]
[[Category:Hilbert space]]