The Real And Complex Number Systems

Proof: Use Cauchy-Schwarz inequality twice, we then have
( n∑
) ⎡ 4 ( n∑
) ⎤ 2
2
a k b k c k = ⎣ a k b k c k
⎦
k=1
k=1
( n∑
) 2 ( n∑
) 2
≤ a 2 kc 2 k b 2 k
k=1
k=1
( n∑
) 2 ( n∑
) ( n∑
≤ a 4 k c 4 k
k=1
k=1 k=1
( n∑
) ( n∑
) 2 ( n∑
= a 4 k b 2 k
k=1 k=1 k=1
b 2 k
c 4 k
) 2
)
.
1.25 Prove that Minkowski’s inequality:
( n∑
k=1
(a k + b k ) 2 ) 1/2
≤
a 2 k) 1/2
+
b 2 k) 1/2
.
( n∑
k=1
( n∑
k=1
This is the triangle inequality ‖a + b‖ ≤ ‖a‖+‖b‖ for n−dimensional vectors,
where a = (a 1 , ..., a n ) , b = (b 1 , ..., b n ) and
‖a‖ =
( n∑
k=1
a 2 k) 1/2
.
Proof: Consider
n∑
n∑ n∑
n∑
(a k + b k ) 2 = a 2 k + b 2 k + 2 a k b k
k=1
k=1
k=1
k=1
(
n∑ n∑
n∑
) 1/2 ( n∑ 1/2
≤ a 2 k + b 2 k + 2 a 2 k bk) 2 by Cauchy-Schwarz inequality
k=1 k=1
k=1
k=1
⎡
1/2 ) ⎤ 1/2
2
= ⎣ ak) 2 +
⎦ .
( n∑
( n∑
k=1
k=1
b 2 k
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