sentences:  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
  where formula_30 and formula_31 are sequences in "Y" that converge to points in formula_21 in the energetic norm.