Let \(F, G \subset \mathcal{H}\) be linear subspaces. An operator \(P\) defined on \(G\) is called ( \(\mathbb{K}\)-)linear if \(P(\alpha f+\beta g)=\alpha P f+\beta P g\) holds for all \(\alpha, \beta \in \mathbb{K}\) and \(f, g \in G\).

(i) Assume that \(F\) is closed and \(P: \mathcal{H} ightarrow F\) is the orthogonal projection. Then show that

\[P^{2}=P \quad \text { and } \quad\langle P g, hangle=\langle g, P hangle \quad \forall g, h \in \mathcal{H}\]

(ii) Show that if \(P: \mathcal{H} ightarrow \mathcal{H}\) is a map satisfying (26.17), then \(P\) is linear and \(P\) is the orthogonal projection onto the closed subspace \(P(\mathcal{H})\).

(iii) Show that if \(P: \mathcal{H} ightarrow \mathcal{H}\) is a linear map satisfying

\[P^{2}=P \quad \text { and } \quad\|P h\| \leqslant\|h\| \quad \forall h \in \mathcal{H}\]

then \(P\) is the orthogonal projection onto the closed subspace \(P(\mathcal{H})\).