Measure, Integration & Real Analysis, 2021a

220 Chapter 8 Hilbert Spaces
The proof above shows that the triangle inequality is an equality if and only if we
have equality in 8.16 and 8.17. Thus we have equality in the triangle inequality if
and only if
8.18 〈 f , g〉 = ‖ f ‖‖g‖.
If one of f , g is a nonnegative multiple of the other, then 8.18 holds, as you should
verify. Conversely, suppose 8.18 holds. Then the condition for equality in the Cauchy–
Schwarz inequality (8.11) implies that one of f , g is a scalar multiple of the other.
Clearly 8.18 forces the scalar in question to be nonnegative, as desired.
Applying the previous result to the inner product space L 2 (μ), where μ is a
measure, gives a new proof of Minkowski’s inequality (7.14) for the case p = 2.
Now we can prove that what we have been calling a norm on an inner product
space is indeed a norm.
8.19 ‖·‖ is a norm
Suppose V is an inner product space and ‖ f ‖ is defined as usual by
‖ f ‖ =
for f ∈ V. Then ‖·‖ is a norm on V.
√
〈 f , f 〉
Proof The definition of an inner product implies that ‖·‖ satisfies the positive definite
requirement for a norm. The homogeneity and triangle inequality requirements
for a norm are satisfied because of 8.6 and 8.15.
The next result has the geometric interpretation
that the sum of the squares
of the lengths of the diagonals of a
parallelogram equals the sum of the
squares of the lengths of the four sides.
8.20 parallelogram equality
Suppose f and g are elements of an inner product space. Then
Proof
We have
‖ f + g‖ 2 + ‖ f − g‖ 2 = 2‖ f ‖ 2 + 2‖g‖ 2 .
‖ f + g‖ 2 + ‖ f − g‖ 2 = 〈 f + g, f + g〉 + 〈 f − g, f − g〉
= ‖ f ‖ 2 + ‖g‖ 2 + 〈 f , g〉 + 〈g, f 〉
+ ‖ f ‖ 2 + ‖g‖ 2 −〈f , g〉−〈g, f 〉
= 2‖ f ‖ 2 + 2‖g‖ 2 ,
as desired.
Measure, Integration & Real Analysis, by Sheldon Axler