Let us take v = c [infinity](r), the space of differentiable functions on the real line, and let d 2 dx2 : c [infinity](r) → c [infinity](r) be the operator that maps a function into its second derivative. show that if ω is a positive constant, then sin √ ωx and cos √ ωx are eigenvectors of d 2 dx2 , and find their corresponding eigenvalues. (note: you don’t have to find the eigenvectors, those are already given, you just need to verify that the given functions are in- deed eigenvectors. also, by doing that computation, you will see what will the corresponding eigenvalues be.) this is a very important operator in differential equation, and the above fact has many important applications in science and engineering. also, in that context, people usually refer to sin √ ωx and cos √ ωx as eigenfunctions.