Although the sampling theorem is well known, its demonstration assumes generalized delta functions and periodic signals, which are not defined in Hilbert spaces, as well as bandlimited signals which not exist in reality because they must have infinite duration.
In this paper, we consider the sampling theorem in Hilbert spaces. We present a simple and complete demonstration of the sampling theorem that uses only signals defined in Hilbert spaces, i.e., we do not use generalized delta functions or periodic signals. We also establish the aliasing effects as a corollary of the theorem.