  In this paper we prove the spectral theorem for quaternionic unbounded normal
operators using the notion of $S$-spectrum. The proof technique consists of
first establishing a spectral theorem for quaternionic bounded normal operators
and then using a transformation which maps a quaternionic unbounded normal
operator to a quaternionic bounded normal operator. With this paper we complete
the foundation of spectral analysis of quaternionic operators. The $S$-spectrum
has been introduced to define the quaternionic functional calculus but it turns
out to be the correct object also for the spectral theorem for quaternionic
normal operators. The fact that the correct notion of spectrum for quaternionic
operators was not previously known has been one of the main obstructions to
fully understanding the spectral theorem in this setting. A prime motivation
for studying the spectral theorem for quaternionic unbounded normal operators
is given by the subclass of unbounded anti-self adjoint quaternionic operators
which play a crucial role in the quaternionic quantum mechanics.
