Chapter 11

Inner Product Spaces

11.1 General Theory

It is assumed here that the field of scalars is either R or C. The usual example of an innerproduct space is Cn or Rn as described earlier. However, there are many other inner productspaces and the topic is of such importance that it seems appropriate to discuss the generaltheory of these spaces.

Definition 11.1.1 A vector space X is said to be a normed linear space if there exists afunction, denoted by |·| : X → [0,∞) which satisfies the following axioms.

1. |x| ≥ 0 for all x ∈ X, and |x| = 0 if and only if x = 0.

2. |ax| = |a| |x| for all a ∈ F.

3. |x+ y| ≤ |x|+ |y| .

This function |·| is called a norm.

The notation ||x|| is also often used. Not all norms are created equal. There are manygeometric properties which they may or may not possess. There is also a concept called aninner product which is discussed next. It turns out that the best norms come from an innerproduct.

Definition 11.1.2 A mapping (·, ·) : V × V → F is called an inner product if it satisfiesthe following axioms.

1. (x, y) = (y, x).

2. (x, x) ≥ 0 for all x ∈ V and equals zero if and only if x = 0.

3. (ax+ by, z) = a (x, z) + b (y, z) whenever a, b ∈ F.

Note that 2 and 3 imply (x, ay + bz) = a(x, y) + b(x, z).Then a norm is given by

(x, x)1/2 ≡ |x| .

It remains to verify this really is a norm.

Definition 11.1.3 A normed linear space in which the norm comes from an inner productas just described is called an inner product space.

Example 11.1.4 Let V = Cn with the inner product given by (x,y) ≡∑n

k=1 xkyk. This isan example of a complex inner product space already discussed.

Example 11.1.5 Let V = Rn,, (x,y) = x · y ≡∑n

j=1 xjyj . This is an example of a realinner product space.

Example 11.1.6 Let V be any finite dimensional vector space and let {v1, · · · , vn} be abasis. Decree that

(vi, vj) ≡ δij ≡

{1 if i = j

0 if i ̸= j

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