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 formula_1 with the inner product formula_2 and the norm formula_3. Let formula_4 be a linear subspace of formula_1 and formula_6 be a strongly monotone symmetric linear operator, that is, a linear operator satisfying 
The energetic inner product is defined as 
formula_14 for all formula_15 in formula_4
and the energetic norm is
formula_17 for all formula_12 in formula_13
The set formula_4 together with the energetic inner product is a pre-Hilbert space. The energetic space formula_21 is defined as the completion of formula_4 in the energetic norm. formula_21 can be considered a subset of the original Hilbert space formula_24 since any Cauchy sequence in the energetic norm is also Cauchy in the norm of formula_1 (this follows from the strong monotonicity property of formula_26). 
The energetic inner product is extended from formula_4 to formula_21 by
 formula_29
where formula_30 and formula_31 are sequences in "Y" that converge to points in formula_21 in the energetic norm.
Energetic extension.
The operator formula_26 admits an energetic extension formula_34 
formula_35
defined on formula_21 with values in the dual space formula_37 that is given by the formula 
formula_38 for all formula_15 in formula_40
Here, formula_41 denotes the duality bracket between formula_37 and formula_43 so formula_44 actually denotes formula_45
If formula_12 and formula_47 are elements in the original subspace formula_48 then
formula_49
by the definition of the energetic inner product. If one views formula_50 which is an element in formula_24 as an element in the dual formula_52 via the Riesz representation theorem, then formula_53 will also be in the dual formula_54 (by the strong monotonicity property of formula_26). Via these identifications, it follows from the above formula that formula_56 In different words, the original operator formula_6 can be viewed as an operator formula_58 and then formula_35 is simply the function extension of formula_26 from formula_4 to formula_40 
An example from physics.
Consider a string whose endpoints are fixed at two points formula_63 formula_64 on the string be formula_65, where formula_66 is a unit vector pointing vertically and formula_67 Let formula_68 be the deflection of the string at the point formula_69 under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is 
 formula_70
and the total potential energy of the string is
 formula_71
The deflection formula_68 minimizing the potential energy will satisfy the differential equation
 formula_73
with boundary conditions
formula_74
To study this equation, consider the space formula_75 that is, the Lp space of all square-integrable functions formula_76 in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
 formula_77
with the norm being given by 
 formula_78
Let formula_4 be the set of all twice continuously differentiable functions formula_76 with the boundary conditions formula_81 Then formula_4 is a linear subspace of formula_83
Consider the operator formula_6 given by the formula
 formula_85
so the deflection satisfies the equation formula_86 Using integration by parts and the boundary conditions, one can see that 
 formula_87
for any formula_12 and formula_47 in formula_13 Therefore, formula_26 is a symmetric linear operator. 
formula_26 is also strongly monotone, since, by the Friedrichs's inequality 
 formula_93
for some formula_94
The energetic space in respect to the operator formula_26 is then the Sobolev space formula_96 We see that the elastic energy of the string which motivated this study is 
 formula_97
so it is half of the energetic inner product of formula_12 with itself. 
To calculate the deflection formula_12 minimizing the total potential energy formula_100 of the string, one writes this problem in the form 
formula_101 for all formula_47 in formula_21.
Next, one usually approximates formula_12 by some formula_105, a function in a finite-dimensional subspace of the true solution space. For example, one might let formula_105 be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation formula_105 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 formula_12 and formula_105, see Céa's lemma.