section_path: Duality
section_name: Duality
paragraph_text: Duality
  
  A linear form is a linear map from a vector space V over a field F to the field of scalars F, viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V, and usually denoted V* or V ′.
  
  If v,..., v is a basis of V (this implies that V is finite-dimensional), then one can define, for i = 1,..., n, a linear map v * such that v *(v) = 1 and v *(v) = 0 if j ≠ i. These linear maps form a basis of V *, called the dual basis of v,..., v. (If V is not finite-dimensional, the v * may be defined similarly; they are linearly independent, but do not form a basis.)
  
  For v in V, the map
  
  is a linear form on V*. This defines the canonical linear map from V into (V *)*, the dual of V*, called the double dual or bidual of V. This canonical map is an isomorphism if V is finite-dimensional, and this allows identifying V with its bidual. (In the infinite dimensional case, the canonical map is injective, but not surjective.)
  
  There is thus a complete symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the bra–ket notation
  
  for denoting f (x).