Note 2
Equality and Inequality
[Parseval’s equality]
Hilbert space X
Complete orthogonal system of X S
S = {x1, x2, …, xn, …} is separable.
Arbitrary x∈X
||x||2 = ∑∞n=1 | (x, xn) |2
[Bessel’s inequality]
Hilbert space X
2 elements of X x, y
When (x, y) = 0, x and y are called orthogonal each other.
Subset S of X does not contain 0 and arbitrary 2 elements are orthogonal each other, S is called orthogonal system.
When each x∈X satisfies ||x|| = 1, S is called normal orthogonal system.
Arbitrary x∈X
∑∞n=1 | (x, xn) |2 ≦||x||2
[Jensen’s inequality]
Positive number p, q
1≦p<q<∞
∑∞n=1 |an| p <∞
∑∞n=1 |an| q <∞
(∑∞n=1 |an| q)1/q ≦(∑∞n=1 |an| p)1/p (0<p≦q)
[Minkowski’s inequality]
Positive number p
1≦p<∞
∑∞n=1 |an| p <∞
∑∞n=1 |bn| p <∞
(∑∞n=1 |an +bn| p)1/p ≦(∑∞n=1 |an| p)1/p + (∑∞n=1 |bn| p)1/p
[Schwarz’s inequality]
Inner product space X
2 elements of X x, y
|(x, y)| ≦
Tokyo May 15, 2008
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