Mean Ergodic Theorem (Hilbert Space)/Lemma

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

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

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

Let $B \subseteq \HH \mu$ the linear subspace defined as:
 * $B := \set {\map {U_T} f - f : f \in \HH }$

Then:
 * $I^\perp \subseteq \overline B$