I am looking for a full--proof to refer to. I have an extracted copy of Boyce & Di Prima's book on differential equations "Elementary Differential Equations", specifically on the section on Sturm-Liouville problems.
In the book, is says "we assume without proof that this problem actually has eigenvalues and eigenfunctions" and adds that "The proof may be found in the references by Sagan(chapter 5) or Birkhoff and Rota(Chapter 10)"
I cannot find which book by either of those authors it is referring to. So I have searched online but while there are copious amounts of proofs for eigenvalues in the context of matrices, none were found for Sturm-Liouville problems.
No one seems to have done it online; is it unbelievably long or complicated? If anyone knows either which book the reference is talking about or, websites which provide the proof, it would be great if I can be directed there.
*I don't know what the best tag for this question is, so please edit/modify if you have better tags to put on
The Sturm-Liouville problem is precisely the linear transformation problem- the only difference is that you need to show that the Sturm-Liouville operator is a self-adjoint linear operator. What is the definition of "adjoint" in the case of second order differential operators?