Theorem (Von Neumann ergodic theorem):
Let be Hilbert space, and let be a unitary operator. Further, let the orthogonal projection onto the space be given by . Then
where the limit is taken with respect to the operator norm on , the space of bounded operators on . Moreover, the inequality
is a valid estimate for the convergence rate.
Proof: Suppose first that and . Then
Further, if we set
If now the sequence is convergent, we see that its limit is indeed contained within . From the respective former consideration, we may hence infer that the sequence does in fact converge to . We are thus reduced to proving the convergence of the sequence in operator norm. Since is Hilbert space, proving that is a Cauchy sequence will be sufficient. But since
for this is the case; the gaps are closed using that
Taking in the next to last computation yields the desired rate of convergence. These computations also reveal the underlying cause of convergence: The sequence becomes more and more uniform, since applying to it does not change it by a large amount.