Measure, Integration & Real Analysis, 2021a

214 Chapter 8 Hilbert Spaces
As we will see, even though the main examples of inner product spaces are L 2 (μ)
spaces, working with the inner product structure is often cleaner and simpler than
working with measures and integrals.
8.3 basic properties of an inner product
Suppose V is an inner product space. Then
(a) 〈0, g〉 = 〈g,0〉 = 0 for every g ∈ V;
(b) 〈 f , g + h〉 = 〈 f , g〉 + 〈 f , h〉 for all f , g, h ∈ V;
(c) 〈 f , αg〉 = α〈 f , g〉 for all α ∈ F and f , g ∈ V.
Proof
(a) For g ∈ V, the function f ↦→ 〈f , g〉 is a linear map from V to F. Because
every linear map takes 0 to 0, wehave〈0, g〉 = 0. Now the conjugate symmetry
property of an inner product implies that
(b) Suppose f , g, h ∈ V. Then
〈g,0〉 = 〈0, g〉 = 0 = 0.
〈 f , g + h〉 = 〈g + h, f 〉 = 〈g, f 〉 + 〈h, f 〉 = 〈g, f 〉 + 〈h, f 〉 = 〈 f , g〉 + 〈 f , h〉.
(c) Suppose α ∈ F and f , g ∈ V. Then
as desired.
〈 f , αg〉 = 〈αg, f 〉 = α〈g, f 〉 = α〈g, f 〉 = α〈 f , g〉,
If F = R, then parts (b) and (c) of 8.3 imply that for f ∈ V, the function
g ↦→ 〈f , g〉 is a linear map from V to R. However, if F = C and f ̸= 0, then
the function g ↦→ 〈f , g〉 is not a linear map from V to C because of the complex
conjugate in part (c) of 8.3.
Cauchy–Schwarz Inequality and Triangle Inequality
Now we can define the norm associated with each inner product. We use the word
norm (which will turn out to be correct) even though it is not yet clear that all the
properties required of a norm are satisfied.
8.4 Definition norm associated with an inner product; ‖·‖
Suppose V is an inner product space. For f ∈ V, define the norm of f , denoted
‖ f ‖,by
√
‖ f ‖ = 〈 f , f 〉.
Measure, Integration & Real Analysis, by Sheldon Axler