10.4. INNER PRODUCT AND NORMED LINEAR SPACES 263

10.4 Inner Product and Normed Linear Spaces10.4.1 The Inner Product in Fn

To do calculus, you must understand what you mean by distance. For functions of onevariable, the distance was provided by the absolute value of the difference of two numbers.This must be generalized to Fn and to more general situations. This is the most familiarsetting for elementary courses. We call it the dot product in calculus and physics but it is acase of something which also works in Cn.

Definition 10.4.1 Let x,y ∈ Fn. Thus x = (x1, · · · ,xn) where each xk ∈ F and a similarformula holding for y. Then the inner product of these two vectors is defined to be

x ·y ≡ (x,y)≡∑j

x jy j ≡ x1y1 + · · ·+ xnyn.

This is also often denoted by (x,y) or as ⟨x,y⟩ and is called an inner product. I will useeither notation.

Notice how you put the conjugate on the entries of the vector, y. It makes no differenceif the vectors happen to be real vectors but with complex vectors you must do it this way2.The reason for this is that when you take the inner product of a vector with itself, you wantto get the square of the length of the vector, a positive number. Placing the conjugate onthe components of y in the above definition assures this will take place. Thus

(x,x) = ∑j

x jx j = ∑j

∣∣x j∣∣2 ≥ 0.

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

(1+ i)2 +22 = 4+2i

and this is not a positive number.The following properties of the inner product follow immediately from the definition

and you should verify each of them.Properties of the inner product:

1. (u,v) = (v,u)

2. If a,b are numbers and u,v,z are vectors then ((au+bv) ,z) = a(u,z)+b(v,z) .

3. (u,u)≥ 0 and it equals 0 if and only if u= 0.

Note this implies (x,αy) = α (x,y) because

(x,αy) = (αy,x) = α (y,x) = α (x,y)

The norm is defined as follows.

Definition 10.4.2 For x ∈ Fn, |x| ≡(

∑nk=1 |xk|2

)1/2= (x,x)1/2

2Sometimes people put the conjugate on the components of the first entry. It doesn’t matter a lot, but it is goodto be consistent. I have chosen to place the conjugate on the components of the second entry.