Uniform Mean Ergodic Theorem

Theorem
Let $\GF \in \set {\R, \C}$.

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\mathbb F$.

Let $U : \HH \to \HH$ be a bounded linear operator such that:
 * $\forall f \in \HH : \norm {\map U f} \le \norm f$

Then for each $f \in \HH$:
 * $\ds \lim_{N - M \mathop \to \infty} \dfrac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^n} f = \map P f$

i.e.
 * $\ds \forall \epsilon \in \R_{>0}: \exists K \in \N: \forall M,N \in \N: N - M \ge K \implies \norm {\frac 1 {N - M} \sum_{n \mathop = M}^{N - 1} \map {U^n} f - \map P f} \le \epsilon$

where:
 * $U^n$ denotes the $n$ times composition of $U$
 * $I := \set {f \in \HH : \map U f = f}$
 * $P : \HH \to I$ denotes the orthogonal projection on $I$