A question about sturm liouville theory

[ShayanJ]

Consider the linear differential equation below:
$Lu=\lambda u$
With L a hermitian operator.
We know that the eigen functions of this equation form a complete set.
So they can build every other function with their linear combination.
But from linearity of the Differential equation,we know that any linear combination of answers is also an answer so it follows that every function is a solution to this equation!
This can easily be shown to be wrong.
What's the problem?

thanks

 

[theorem4.5.9]

The same could be said for most linear transformations on finite dimensional vector spaces. The main problem is that different eigenvectors typically have different eigenvalues, so sums of eigenvectors are not usually eigenvectors (unless they all have the same eigenvalue).

 

[ShayanJ]

Oh,yes!
You mean we have:
$Lu_1=\lambda_1 u_1$
but
$Lu_2=\lambda_2 u_2$
Yeah.
I understand now
thanks