sentences:  The "Frobenius norm" defined by
  
  is self-dual, i.e., its dual norm is formula_27
  
  The "spectral norm", a special case of the "induced norm" when formula_28, is defined by the maximum singular values of a matrix, i.e.,
  
  has the nuclear norm as its dual norm, which is defined by 
  
  for any matrix formula_31 where formula_32 denote the singular values.More generally, let formula_1 and formula_34 be topological vector spaces, and formula_35 be the collection of all bounded linear mappings (or "operators") of formula_1 into formula_34.In the case where formula_1 and formula_34 are normed vector spaces, formula_35 can be normed in a natural way.