A hermitian form H(,) on a complex vector space satisfies:

  • H(x,y) is linear in x for all y
  • H(x,y) = H(y,x)* for all vectors x and y

Given such a from on a complex vector space, it defines a natural way to associate the space with its dual. Knowing this, we call the dual map to a map on the space its hermitian adjoint. If a map is equal to its hermitian adjoint, it is said to be hermitian.