  A simple proof is provided to show that any bounded normal operator on a real
Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure
theorem for invertible skew-symmetric operators, which is analogous to the
finite dimensional situation is also proved using elementary techniques. The
second result is used to establish the main theorem of this article, which is a
generalization of Williamson's normal form for bounded positive operators on
infinite dimensional separable Hilbert spaces. This has applications in the
study of infinite mode Gaussian states.
