Bessel's inequality - Alchetron, The Free Social Encyclopedia

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence.

Let H be a Hilbert space, and suppose that e 1 , e 2 , . . . is an orthonormal sequence in H . Then, for any x in H one has

∑ k = 1 ∞ | ⟨ x , e k ⟩ | 2 ≤ ∥ x ∥ 2

where 〈•,•〉 denotes the inner product in the Hilbert space H . If we define the infinite sum

x ′ = ∑ k = 1 ∞ ⟨ x , e k ⟩ e k ,

consisting of 'infinite sum' of vector resolute x in direction e k , Bessel's inequality tells us that this series converges. One can think of it that there exists x ′ ∈ H which can be described in terms of potential basis e 1 , e 2 , . . . .

For a complete orthonormal sequence (that is, for an orthonormal sequence which is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently x ′ with x ).

Bessel's inequality follows from the identity:

0 ≤ ∥ x − ∑ k = 1 n ⟨ x , e k ⟩ e k ∥ 2 = ∥ x ∥ 2 − 2 ∑ k = 1 n | ⟨ x , e k ⟩ | 2 + ∑ k = 1 n | ⟨ x , e k ⟩ | 2 = ∥ x ∥ 2 − ∑ k = 1 n | ⟨ x , e k ⟩ | 2 ,

which holds for any natural n.

References

Bessel's inequality Wikipedia

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