Fredholm's alternative & L2 convergence

[eousseu]

Hello everyone,
I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he
gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem.
Assume that we are working on L2 with the following inner product

(f,g) = ∫f*g*m dx

where m(x)>0.At some point he concludes that |u(x)| ≤

∑((∫fundx)/(δ(un,un))*un (sum from 1 to inf)

Then assuming that u_n are normalized eigenvectors he uses the Cauchy-Schwarz inequality and parseval's identity to conclude the following¨

||u||2 ≤ (1/δ2)∑|(∫fundx)|2

The problem is that when i try to use the same logic in order to find this bound i end up having an infinite sum of ones on the second member of the inequality.
Thanks!.

 

[Infrared]

It's rather hard for me to answer this question because of how many terms you left undefined. What are the $u_n$? Is some function expressed as a linear combination of them? What are they eigenvectors of (and is this relevant)? I'm assuming they're orthogonal? What is $\delta(u_n,u_n)$? What is $\delta_2,$ or is that supposed to be $\delta^2?$ Etc.

Also please consider writing your post in latex to make it easier to read.

 

[eousseu]

u(x) is the solution of a non homogeneous elliptic problem(analogous to the sturm -liouville d.e.) where f is the nonhomogeneous part and δ is the minimum of the difference between the eigenvalues and the coefficient of the term u(x) (assuming it is different from the eigenvalues of the Laplace operator).I did not present any details about the problem because i thought they are irrelevant(although they might not be). I just need to get from one line of the proof to the next. By δ2 i actually meant δ^2. Sorry for being unclear on this. Thanks.