(Spectrum of compact operator) Let
be a Hilbert space,
and
is a linear compact operator. Then

1.
.

2.

3. Either
is finite or it is a sequence converging to 0.

Proposition

(Eigenvalues of compact
symmetric operator) Let
be a separable Hilbert space and
is a linear compact operator. Then there exists a countable orthonormal basis
of
consisting of eigenvectors of
.