Not "complement", "complex conjugate". The definition of an inner product on a vector space over the complex numbers requires that . To take a simple example, if u= (a+bi, c+di) is a vector in , pairs of complex numbers, and v= (e+fi,g+hi) is another, then the inner product on is . Inner products on vector spaces over the complex numbers are defined that way to guarentee that thenorm, will be a positive real number. If, for example, v= (i, i) and we do NOT require the complex conjugate, we would have <v, v>= i(i)+ i(i)= -1 and then . That's a problem since we want to use norm to "compare" the size of vectors and the complex numbers is not an ordered field. Even worse, if v= (1, i) then <v, v>= 1(1)+ i(i)= 1-1= 0 so ||v||= 0 even though v is not the 0 vector.

Using the correct inner product, if v= (i, i) then [tex]<v, v>= i(-i)+ i(-i)= 1+ 1= 2[tex] and . And if v= (1, i), and .