NORMED SPACE OF

4 S. EKARIANI, H. GUNAWAN, M. IDRIS
By Minkowski’s inequality, we have
Hence we obtain
⎛
∣ ∣
⎝ 1 ∑ ∑ ∑ ∑ ∣∣∣∣∣ ∣∣∣∣∣ η 1j1 η 1j2 η 1j3
ξ k η 2j1 η 2j2 η 2j3
3!
k j 1 j 2 j 3 η 3j1 η 3j2 η 3j3
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
⎛
∣ ∣
≤ ⎝ 1 ∑ ∑ ∑ ∑ ∣∣∣∣∣ ∣∣∣∣∣ η 2j2 η 2k η 2j3
η 1j1 ξ j2 ξ k ξ j3
3!
k j 1 j 2 j 3 η 3j2 η 3k η 3j3
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
⎛
∣ ∣
⎝ 1 ∑ ∑ ∑ ∑ ∣∣∣∣∣ ∣∣∣∣∣ η 2j1 η 2k η 2j3
η 1j2 η 3j1 η 3k η 3j3
3!
k j 1 j 2 j 3 ξ j1 ξ k ξ j3
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
⎛
∣ ∣
⎝ 1 ∑ ∑ ∑ ∑ ∣∣∣∣∣ ∣∣∣∣∣ η 2j1 η 2k η 2j2
η 1j3 ξ j1 ξ k ξ j2
3!
k j 1 j 2 j 3 η 3j1 η 3k η 3j2
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
⎛
∣ ∣
⎝ 1 ∑ ∑ ∑ ∑ ∣∣∣∣∣ ∣∣∣∣∣ η 1j1 η 1j3 η 1j2
η 2k η 3j1 η 3j3 η 3j2
3!
k j 1 j 2 j 3 ξ j1 ξ j3 ξ j2
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
⎛
∣ ∣ ∣∣∣∣∣ ∣∣∣∣∣
⎝ 1 ∑ ∑ ∑ ∑ η 1j1 η 1j3 η 1j2
η 3k ξ j1 ξ j3 ξ j2
3!
k j 1 j 2 j 3 η 2j1 η 2j3 η 2j2
∣∣ ∣∣∣∣∣ ∣∣∣∣∣ p⎞
⎠
= 3 ∥y 1 ∥ p ∥x, y 2 , y 3 ∥ p + ∥y 2 ∥ p ∥x, y 1 , y 3 ∥ p + ∥y 3 ∥ p ∥x, y 1 , y 2 ∥ p .
∥x∥ p ∥y 1 , y 2 , y 3 ∥ p ≤ 3∥y 1 ∥ p ∥x, y 2 , y 3 ∥ p + ∥y 2 ∥ p ∥x, y 1 , y 3 ∥ p + ∥y 3 ∥ p ∥x, y 1 , y 2 ∥ p ,
1
p
1
p
1
p
1
p
1
p
1
p
+
+
+
+
as desired.
□
Proposition 2.5. Let {a 1 , . . . , a n } be a linearly independent set on l p . Then the
norm ∥ ⋅ ∥ ∗ p defined in Proposotion 2.3 is equivalent to the usual norm ∥ ⋅ ∥ p on l p .
Precisely, we have
n∥a 1 , . . . , a n ∥ p
(2n − 1) [∥a 1 ∥ p + ⋅ ⋅ ⋅ + ∥a n ∥ p ] ∥x∥ p ≤ ∥x∥ ∗ p
for every x ∈ l p .
⎡
≤ (n!) 1− 1 p ⎣
∑
{i 2,...,i n}⊂{1,...,n}
⎤
∥a i2 ∥ p p ⋅ ⋅ ⋅ ∥a in ∥ p ⎦
p
1
p
∥x∥ p
Proof. For any x ∈ l p and any subset {i 2 , . . . , i n } of {1, 2, . . . , n}, we observe that
∥x, a i2 , . . . , a in ∥ p ≤ (n!) 1−(1/p) ∥x∥ p ∥a i2 ∥ p ⋅ ⋅ ⋅ ∥a in ∥ p .