sentences:  An example of this will be given later in the article.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 
  and the energetic norm is
  
  The set formula_4 together with the energetic inner product is a pre-Hilbert space.