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I lecture today we talked about Bessel's inequality and it's use in showing convergence of orthonormal sequences in a Hilbert space.

$\sum_{n=1}^\infty \lvert <x,e_n>\rvert^2 \leqslant \lVert x\rVert^2 \implies e_n\to 0$ (weakly)

I am not sure how we get from Bessel's to the implication of convergence? I get that we know that each $<x,e_n>$ is positive, and that it is bounded by $\lVert x\rVert^2$, but I am not sure what I am missing...

Thanks!

user3784030
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  • In calculus you should have learned that an absolutely convergent sum is convergent. And you should have learned that $\lim_{n} a_{n}=0$ if $\sum_{n=1}^{\infty}a_{n}$ converges converges conditionally or absolutely. – Disintegrating By Parts Sep 11 '14 at 05:31

1 Answers1

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Just notice that

$$ \sum_{n=1}^\infty \lvert <x,e_n>\rvert^2 $$

is a convergent series which implies

$$ \lim_{n\to \infty} \lvert <x,e_n>\rvert^2 = 0. $$

Note: We used the fact

If $\sum_n a_n $ converges then $\lim_{n\to \infty}a_n=0$.

Mhenni Benghorbal
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