# Functional analysis, ortho basis, weakly convergent

1. Dec 13, 2015

### Fellowroot

1. The problem statement, all variables and given/known data

This is a problem from Haim Brezis's functional analysis book.
2. Relevant equations

3. The attempt at a solution

I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on.

We know every hilbert space has an orthonormal basis.

I also need to know the difference between l^2 and L^2 spaces. I know that this might not be much of a solution, but I'm really stuck.

Last edited by a moderator: Dec 13, 2015
2. Dec 14, 2015

### Samy_A

You could start by stating clearly what weak convergence implies (here in the context of a Hilbert space).

Once you have a clear understanding of weak convergence, 1. and 2. shouldn't be too difficult.
As ($e_n)$ is an orthornormal basis of the Hilbert space H, you can use $\forall x \in H: x= \sum_{i=1}^\infty <x,e_i>e_i$ (series converging in the norm of the Hilbert space).
And also $\forall x \in H:|x|²=\sum_{i=1}^\infty |<x,e_i>|²$.

Last edited: Dec 14, 2015
3. Dec 14, 2015

### Fellowroot

Thanks.

I think I now have the answer to part 1 because you can use an idea from Parseval which automically shows that en weakly converges to zero. But I'm still not sure about what to do with an and un. I also think the boundedness of an plays a role.

For part 2 I know that this is showing strongly convergence, but I don't really get how to show abs(un). Do I just pretend that the summation is really a sequence?

4. Dec 14, 2015

### Samy_A

Yes, that is correct.
You can compute an upper bound for $|u_n|$, making use of the fact that $(a_n)$ is bounded, and $(e_n)$ orthonormal.
You can use that, in general, you have:
$|\sum_{i=1}^n b_ie_i|²=\sum_{i=1}^n |b_i|²$ (where $(b_i)$ are any numbers), because $(e_n)$ is orthonormal.