proof of Bessel inequality
Let
rn=x-n∑k=1⟨x,ek⟩⋅ek. |
Then for j=1,…,n,
⟨rn,ej⟩ | =⟨x,ej⟩-n∑k=1⟨⟨x,ek⟩⋅ek,ej⟩ | (1) | |||
=⟨x,ej⟩-⟨x,ej⟩⟨ej,ej⟩=0 | (2) |
so e1,…,en,rn is an orthogonal series.
Computing norms, we see that
∥x∥2=∥rn+n∑k=1⟨x,ek⟩⋅ek∥2=∥rn∥2+n∑k=1|⟨x,ek⟩|2≥n∑k=1|⟨x,ek⟩|2. |
So the series
∞∑k=1|⟨x,ek⟩|2 |
converges and is bounded by ∥x∥2, as required.
Title | proof of Bessel inequality |
---|---|
Canonical name | ProofOfBesselInequality |
Date of creation | 2013-03-22 12:46:41 |
Last modified on | 2013-03-22 12:46:41 |
Owner | ariels (338) |
Last modified by | ariels (338) |
Numerical id | 4 |
Author | ariels (338) |
Entry type | Proof |
Classification | msc 46C05 |