sentences:  Assuming that the deflection is small, the elastic energy of the string is 
  
  and the total potential energy of the string is
  
  The deflection formula_68 minimizing the potential energy will satisfy the differential equation
  
  with boundary conditions
  
  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
  
  with the norm being given by 
  
  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
  
  so the deflection satisfies the equation formula_86 Using integration by parts and the boundary conditions, one can see that 
  
  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 
  
  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 
  
  so it is half of the energetic inner product of formula_12 with itself.