In a lecture of Nuclear Physics in which my professor was reviewing some result of functional analysis he said that a condition which can be used to tell if a orthonormal set is complete is:
$$ \sum_n W_n^*(\xi')W_n(\xi) = \delta(\xi'-\xi) $$
I can't find references for this equation, is a sufficient or a necessary condition? How can it be proven?
EDIT: In the same lecture he said that the coefficient $a_n$ of the expansion of a function $f(\xi)$ in order to obtain: $$ f(\xi) = \sum_n a_n W_n(\xi) $$
can be calculated by the formula: $$ a_n = \int W_n^* f W_n d\xi $$
But if i remember correctly my (too brief) course of introduction to functional analysis the teacher tell me that $a_n$ is given by the inner product and so the coefficient should be calculated by: $$ a_n = \int f^* W_n d\xi $$
Which one is correct? What is the difference?
Your second question answers your first question. As this is coming from a physics class I am going to blithely ignore all issues of convergence, etc. Once we have
$$a_n = \int f(\xi) W_n^{\ast}(\xi) \, d \xi$$
we get
$$f(\xi) = \sum \left( \int f(\xi') W_n^{\ast}(\xi') \, d \xi' \right) W_n(\xi) = \int f(\xi') \sum W_n^{\ast}(\xi') W_n(\xi) \, d \xi'.$$
On the other hand, we also have
$$f(\xi) = \int f(\xi') \delta(\xi' - \xi) \, d \xi'.$$
It is perhaps easier to first understand the analogous finite-dimensional statement, which becomes a statement about how to write the identity matrix in terms of an orthonormal basis of a finite-dimensional Hilbert space.