Measure, Integration & Real Analysis, 2021a

Section 8A Inner Product Spaces 213
8.2 Example inner product spaces
• For n ∈ Z + , define an inner product on F n by
〈(a 1 ,...,a n ), (b 1 ,...,b n )〉 = a 1 b 1 + ···+ a n b n
for (a 1 ,...,a n ), (b 1 ,...,b n ) ∈ F n . When thinking of F n as an inner product
space, we always mean this inner product unless the context indicates some other
inner product.
• Define an inner product on l 2 by
〈(a 1 , a 2 ,...), (b 1 , b 2 ,...)〉 =
∞
∑ a k b k
k=1
for (a 1 , a 2 ,...), (b 1 , b 2 ,...) ∈ l 2 . Hölder’s inequality (7.9), as applied to counting
measure on Z + and taking p = 2, implies that the infinite sum above
converges absolutely and hence converges to an element of F. When thinking
of l 2 as an inner product space, we always mean this inner product unless the
context indicates some other inner product.
• Define an inner product on C([0, 1]), which is the vector space of continuous
functions from [0, 1] to F,by
〈 f , g〉 =
∫ 1
0
f g
for f , g ∈ C([0, 1]). The definiteness requirement for an inner product is
satisfied because if f : [0, 1] → F is a continuous function such that ∫ 1
0 | f |2 = 0,
then the function f is identically 0.
• Suppose (X, S, μ) is a measure space. Define an inner product on L 2 (μ) by
∫
〈 f , g〉 =
f g dμ
for f , g ∈ L 2 (μ). Hölder’s inequality (7.9) with p = 2 implies that the integral
above makes sense as an element of F. When thinking of L 2 (μ) as an inner
product space, we always mean this inner product unless the context indicates
some other inner product.
Here we use L 2 (μ) rather than L 2 (μ) because the definiteness requirement fails
on L 2 (μ) if there exist nonempty sets E ∈Ssuch that μ(E) =0 (consider
〈χ E
, χ E
〉 to see the problem).
The first two bullet points in this example are special cases of L 2 (μ), taking μ to
be counting measure on either {1, . . . , n} or Z + .
Measure, Integration & Real Analysis, by Sheldon Axler