Functional analysis, ortho basis, weakly convergent

[Fellowroot]

Homework Statement

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

Homework Equations

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.

 

[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 $\forall x \in H:|x|²=\sum_{i=1}^\infty |<x,e_i>|²$.

 

[Fellowroot]

Thanks.

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

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

 

[Samy_A]

I think I now have the answer to part 1 because you can use an idea from Parseval which automically shows that e_n weakly converges to zero.

Yes, that is correct.

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

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.