Saturday 28 April 2018

Functional Analysis Note 2 Equality and Inequality


Note 2

Equality and Inequality 


[Parseval’s equality]
Hilbert space    X
Complete orthogonal system of X     S
S = {x1x2, …, xn, …} is separable.
Arbitrary xX
||x||2 = ∑n=1 | (xxn) |2

[Bessel’s inequality]
Hilbert space     X
2 elements of X     xy
When (xy) = 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 xX satisfies ||x|| = 1, S is called normal orthogonal system.
Arbitrary xX
n=1 | (xxn) |||x||2

[Jensen’s inequality]
Positive number     pq
1p<q<
n=1 |anp <
n=1 |anq <
(∑n=1 |anq)1/(∑n=1 |anp)1/p  (0<pq)

[Minkowski’s inequality]
Positive number     p
1p<
n=1 |anp <
n=1 |bnp <
(∑n=1 |a+bnp)1/p (∑n=1 |anp)1/p + (∑n=1 |bnp)1/p

[Schwarz’s inequality]
Inner product space     X
2 elements of X     xy
|(xy)|  

Tokyo May 15, 2008

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