sentences:  The operator formula_26 admits an energetic extension formula_34 
  
  defined on formula_21 with values in the dual space formula_37 that is given by the formula 
  
  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
  
  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 
  
  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.