404 CHAPTER 17. INNER PRODUCT SPACES
Example 17.1.3 Let V be any complex vector space and let {v1, · · · ,vn} be a basis. Decreethat ⟨
vi,v j⟩= δ i j.
Then define ⟨n
∑j=1
c jv j,n
∑k=1
dkvk
⟩≡∑
j,kc jdk
⟨v j,vk
⟩=
n
∑k=1
ckdk
This makes the complex vector space into an inner product space.
Example 17.1.4 Let V consist of sequences a = {ak}∞
k=1 , ak ∈ C, with the property that
∞
∑k=1|ak|2 < ∞
and the inner product is then defined as
⟨a,b⟩ ≡∞
∑k=1
akbk
All of the axioms of the inner product are obvious for this example except the mostbasic one which says that the inner product has values in C. Why does the above sum evenconverge? It converges from a comparison test.
∣∣akbk∣∣≤ |ak|2
2+|bk|2
2
and by assumption,∞
∑k=1
(|ak|2
2+|bk|2
2
)< ∞
and therefore, the given sum which defines the inner product is absolutely convergent.Therefore, thanks to completeness of C this sum also converges. This fact should be famil-iar to anyone who has had a calculus class in the context that the sequences are real valued.The case where they are complex valued follows right away from a consideration of realand imaginary parts.
By far the most important example of an inner product space is L2 (Ω), the space ofLebesgue measurable square integrable functions defined on Ω. However, this is a book onalgebra, not analysis, so this example will be ignored.
17.1.1 The Cauchy Schwarz Inequality And NormsThe most fundamental theorem relative to inner products is the Cauchy Schwarz inequality.
Theorem 17.1.5 (Cauchy Schwarz)The following inequality holds for x and y ∈ V, an in-ner product space.
|⟨x,y⟩| ≤ ⟨x,x⟩1/2 ⟨y,y⟩1/2 (17.1)
Equality holds in this inequality if and only if one vector is a multiple of the other.