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5.3 Inner-Product Spaces 291
guarantees that x y + x z = x y + z . Now prove that x αy = α x y
for all real α. This is valid for integer values of α by the result just established,
and it holds when α is rational because if β and γ are integers, then
β β β
2
γ x y = γx βy = βγ x y =⇒ x y = x y .
γ γ γ
Because x + αy and x − αy are continuous functions of α (Exercise
5.1.7), equation (5.3.8) insures that x αy is a continuous function of α. There-
fore, if α is irrational, and if {α n } is a sequence of rational numbers such that
α n → α, then x α n y → x αy and x α n y = α n x y → α x y , so
x αy = α x y .
Example 5.3.4
n
We already know that the euclidean vector norm on C is generated by the stan-
dard inner product, so the previous theorem guarantees that the parallelogram
identity must hold for the 2-norm. This is easily corroborated by observing that
2 2 ∗ ∗
x + y + x − y =(x + y) (x + y)+(x − y) (x − y)
2 2
2 2
=2 (x x + y y)=2( x + y ).
∗
∗
2 2
The parallelogram identity is so named because it expresses the fact that the
sum of the squares of the diagonals in a parallelogram is twice the sum of the
squares of the sides. See the following diagram.
x + y
y ||x - y|| ||x + y||
||y||
||x|| x
Example 5.3.5
Problem: Except for the euclidean norm, is any other vector p-norm generated
by an inner product?
Solution: No, because the parallelogram identity (5.3.7) doesn’t hold when
2 2 2 2
p
=2. To see that x + y + x − y =2 x + y p is not valid for
p
p
p
n
all x, y ∈C when p
=2, consider x = e 1 and y = e 2 . It’s apparent that
2 2/p 2
e 1 + e 2 =2 = e 1 − e 2 , so
p p
2 2 (p+2)/p 2 2
e 1 + e 2 + e 1 − e 2 =2 and 2 e 1 + e 2 =4.
p p p p