Dimostrazione.
$f$ è unitario se e solo se $\, \forall \mathbf{v},\mathbf{w} \in \mathbf{E}:$
$\mathbf{v} \cdot \mathbf{w} = f(\mathbf{v}) \cdot f(\mathbf{w})=\mathbf{v} \cdot f^{*}(f(\mathbf{w}))= \mathbf{v} \cdot (f^{*}\circ f)(\mathbf{w})$.
Quindi, dato $\mathbf{w} \in \mathbf{E}$, si ha che $\forall \mathbf{v} \in \mathbf{E}$:
$\mathbf{v} \cdot (\mathbf{w}-(f^{*} \circ f)\mathbf{w})=0$,
quindi $ \mathbf{w}=(f^{*} \circ f)(\mathbf{w}), \quad \forall \mathbf{w} \in \mathbf{E} \,\,$ e cioè $f^{*} \circ f=1_{\mathbf{E}}$
c.v.d.