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  with the inner product  and the norm . Let  be a linear subspace of  and  be a strongly monotone symmetric linear operator, that is, a linear operator satisfying 

  for all  in 
  for some constant  and all  in 

The energetic inner product is defined as 
 for all  in 
and the energetic norm is
 for all  in 

The set  together with the energetic inner product is a pre-Hilbert space. The energetic space  is defined as the completion of  in the energetic norm.  can be considered a subset of the original Hilbert space  since any Cauchy sequence in the energetic norm is also Cauchy in the norm of  (this follows from the strong monotonicity property of ). 

The energetic inner product is extended from  to  by
 
where  and  are sequences in Y that converge to points in  in the energetic norm.

Energetic extension
The operator  admits an energetic extension  

defined on  with values in the dual space  that is given by the formula 

 for all  in 

Here,  denotes the duality bracket between  and  so  actually denotes  

If  and  are elements in the original subspace  then

by the definition of the energetic inner product.  If one views  which is an element in  as an element in the dual  via the Riesz representation theorem, then  will also be in the dual  (by the strong monotonicity property of ). Via these identifications, it follows from the above formula that  In different words, the original operator  can be viewed as an operator  and then  is simply the function extension of  from  to

An example from physics

Consider a string whose endpoints are fixed at two points  on the real line   (here viewed as a horizontal line). Let the vertical outer force density at each point   on the string be , where   is a unit vector pointing vertically and  Let  be the deflection of the string at the point  under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is 

 

and the total potential energy of the string is

 

The deflection  minimizing the potential energy will satisfy the differential equation

 

with boundary conditions

To study this equation, consider the space  that is, the Lp space of all square-integrable functions  in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

 

with the norm being given by 

 

Let  be the set of all twice continuously differentiable functions  with the boundary conditions  Then  is a linear subspace of 

Consider the operator  given by the formula

 

so the deflection satisfies the equation  Using  integration by parts and the boundary conditions, one can see that 

 

for any  and  in  Therefore,  is a symmetric linear operator. 

 is also strongly monotone, since, by the Friedrichs's inequality 

 

for some 

The energetic space in respect to the operator  is then the Sobolev space  We see that the elastic energy of the string which motivated this study is 

 

so it is half of the energetic inner product of  with itself. 

To calculate the deflection  minimizing the total potential energy  of the string, one writes this problem in the form 

 for all  in .

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

See also
 Inner product space
 Positive-definite kernel

References

Functional analysis
Hilbert spaces