Page 118 - Linear Algebra Done Right

Norms
105
6.9
can be used to show
u + v ≤ u + v .
6.10 Triangle Inequality: If u, v ∈ V, then The triangle inequality
that the shortest path
between two points is a
This inequality is an equality if and only if one of u, v is a nonnegative straight line segment.
multiple of the other.
Proof: Let u, v ∈ V. Then
2
u + v = u + v, u + v
= u, u + v, v + u, v + v, u

= u, u + v, v + u, v + u, v
2 2
= u + v + 2Re u, v
2
2
6.11 ≤ u + v + 2| u, v |
2
2
6.12 ≤ u + v + 2 u v
2
= ( u + v ) ,
where 6.12 follows from the Cauchy-Schwarz inequality (6.6). Taking
square roots of both sides of the inequality above gives the triangle
inequality 6.10.
The proof above shows that the triangle inequality 6.10 is an equality
if and only if we have equality in 6.11 and 6.12. Thus we have equality
in the triangle inequality 6.10 if and only if

6.13 u, v = u v .

If one of u, v is a nonnegative multiple of the other, then 6.13 holds, as
you should verify. Conversely, suppose 6.13 holds. Then the condition
for equality in the Cauchy-Schwarz inequality (6.6) implies that one of
u, v must be a scalar multiple of the other. Clearly 6.13 forces the
scalar in question to be nonnegative, as desired.

The next result is called the parallelogram equality because of its
geometric interpretation: in any parallelogram, the sum of the squares
of the lengths of the diagonals equals the sum of the squares of the
lengths of the four sides.