Functional Analysis Note 2 Equality and Inequality / May 15, 2008

Functional Analysis

Note 2

Equality and Inequality 

TANAKA Akio

[Parseval’s equality]

Hilbert space    X

Complete orthogonal system of X     S

S = {x₁, x₂, …, x_n, …} is separable.

Arbitrary x∈X

||x||² = ∑^∞_n=1 | (x, x_n) |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, x_n) |² ≦||x||²

[Jensen’s inequality]

Positive number　　　　　p, q

1≦p<q<∞

∑^∞_n=1 |a_n| ^p <∞

∑^∞_n=1 |a_n| ^q <∞

(∑^∞_n=1 |a_n| ^q)^1/q ≦(∑^∞_n=1 |a_n| ^p)^1/p  (0<p≦q)

[Minkowski’s inequality]

Positive number　　　　　p

1≦p<∞

∑^∞_n=1 |a_n| ^p <∞

∑^∞_n=1 |b_n| ^p <∞

(∑^∞_n=1 |a_n +b_n| ^p)^1/p ≦(∑^∞_n=1 |a_n| ^p)^1/p + (∑^∞_n=1 |b_n| ^p)^1/p

[Schwarz’s inequality]

Inner product space     X

2 elements of X     x, y

|(x, y)| ≦ 

Tokyo May 15, 2008

Sekinan Research Field of Language