Page 117 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
104
In 1821 the French
mathematician
| u, v | ≤ u v .
6.7
Augustin-Louis Cauchy 6.6 Cauchy-Schwarz Inequality: If u, v ∈ V, then
showed that this This inequality is an equality if and only if one of u, v is a scalar mul-
inequality holds for the tiple of the other.
inner product defined
by 6.1. In 1886 the Proof: Let u, v ∈ V.If v = 0, then both sides of 6.7 equal 0 and
German mathematician the desired inequality holds. Thus we can assume that v = 0. Consider
Herman Schwarz the orthogonal decomposition
showed that this u, v
u = v + w,
inequality holds for the v 2
inner product defined where w is orthogonal to v (here w equals the second term on the right
by 6.2.
side of 6.5). By the Pythagorean theorem,
2
2 u, v 2
u = v + w
2
v
2
| u, v | 2
= 2 + w
v
2
| u, v |
6.8 ≥ .
v 2
2
Multiplying both sides of this inequality by v and then taking square
roots gives the Cauchy-Schwarz inequality 6.7.
Looking at the proof of the Cauchy-Schwarz inequality, note that 6.7
is an equality if and only if 6.8 is an equality. Obviously this happens if
and only if w = 0. But w = 0 if and only if u is a multiple of v (see 6.5).
Thus the Cauchy-Schwarz inequality is an equality if and only if u is a
scalar multiple of v or v is a scalar multiple of u (or both; the phrasing
has been chosen to cover cases in which either u or v equals 0).
The next result is called the triangle inequality because of its geo-
metric interpretation that the length of any side of a triangle is less
than the sum of the lengths of the other two sides.

u + v
v

u
The triangle inequality