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mu = ∑ c m n = ∑ c δ = c m (7.29)
n
n mn,
n n
Next, using the Dirac notation, we present the proofs of two key theorems
of vector algebra: the Cauchy-Schwartz inequality and the triangle inequality.

7.4.1 Cauchy-Schwartz Inequality
Let u and v be any non-zero vectors; then:

2
uv ≤ u u v v (7.30)

PROOF Let ε = ±1, (ε = 1); then
2
ε = 1 if uv ≥ 0

uv = ε uv such that  (7.31)
ε =−1 if uv ≤ 0

Now, consider the ket εutv+ ; its norm is always non-negative. Computing
this norm square, we obtain:
εu tv+ εu tv+ = ε u u + εt u v + ε t v u + t v v
2
2
= uu + 2 εt u v + t v v (7.32)
2
= uu + 2 t u v + t v v
2

The RHS of this quantity is a positive quadratic polynomial in t, and can be
written in the standard form:

at + bt + c ≥ 0 (7.33)
2
The non-negativity of this quadratic polynomial means that it can have at most
one real root. This means that the descriminant must satisfy the inequality:

b – 4ac ≤ 0 (7.34)
2
Replacing a, b, c by their values from Eq. (7.32), we obtain:

2
4 uv − 4 u u v v ≤ 0 (7.35)

2
⇒ uv ≤ u u v v (7.36)

© 2001 by CRC Press LLC