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proof of Bessel inequality


Let

rn=x-nk=1x,ekek.

Then for j=1,,n,

rn,ej =x,ej-nk=1x,ekek,ej (1)
=x,ej-x,ejej,ej=0 (2)

so e1,,en,rn is an orthogonal series.

Computing norms, we see that

x2=rn+nk=1x,ekek2=rn2+nk=1|x,ek|2nk=1|x,ek|2.

So the series

k=1|x,ek|2

converges and is bounded by x2, 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