It is necessary to give a generalization of the dot product for vectors in ℂ^{n}. This definition reduces to the usual one in the case the components of the vector are real.
Definition 3.2.12 Let x,y ∈ ℂ^{n}. Thus x =

Notice how you put the conjugate on the entries of the vector y. It makes no difference if the vectors happen to be real vectors but with complex vectors you must do it this way. The reason for this is that when you take the dot product of a vector with itself, you want to get the square of the length of the vector, a positive number. Placing the conjugate on the components of y in the above definition assures this will take place. Thus

If you didn’t place a conjugate as in the above definition, things wouldn’t work out correctly. For example,

and this is not a positive number.
The following properties of the dot product follow immediately from the definition and you should verify each of them.
Properties of the dot product:
The norm is defined in the usual way.
Definition 3.2.13 For x ∈ ℂ^{n},

As in the case of ℝ^{n}, the Cauchy Schwarz inequality is of fundamental importance. First here is a simple lemma.
Lemma 3.2.14 If z ∈ ℂ there exists θ ∈ ℂ such that θz =
Proof: Let θ = 1 if z = 0 and otherwise, let θ =
Proof: Let θ ∈ ℂ such that

which yields the Cauchy Schwarz inequality. ■
By analogy to the case of ℝ^{n}, length or magnitude of vectors in ℂ^{n} can be defined.
Proof: The first two claims are left as exercises. To establish the third, you use the same argument which was used in ℝ^{n}.
Definition 3.2.18 Suppose you have a vector space, V and for z,w ∈ V and α a scalar a norm is a way of measuring distance or magnitude which satisfies the properties 3.16  3.18. Thus a norm is something which does the following.
 (3.19) 
 (3.20) 
 (3.21) 
Here is understood that for all z ∈ V,