Mean Ergodic Theorem (Hilbert Space)/Lemma

Lemma
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:
 * $\forall f \in \HH : \norm {\map U f} \le \norm f$

Let $B \subseteq \HH$ be the linear subspace defined as:
 * $B := \set {\map U h - h : h \in \HH }$

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

Proof
It suffice to show $B^\perp \subseteq I$.

Indeed, then it follows:

Recall that the adjoint $U^\ast$ of $U$ exists by Existence and Uniqueness of Adjoint.

Let $f \in B^\perp$.

Then $\map U f = f$, as:

Therefore $\map U f = f$, as: