Exercise 12.2: Projections in a Hilbert space
(a)
By definition of the sequence \(( g_n )_n\), we have \(\| \tfrac{g_n + g_m}{2} - f \| \leq \tfrac{1}{2} \| g_n - f \| + \tfrac{1}{2} \| g_m - f \| \to p_\star\). Rewriting \begin{align} 4 \| \tfrac{g_n + g_m}{2} - f \|^2 = \| g_n - f \|^2 + \| g_m - f \|^2 + 2 \langle g_n - f , g_m - f \rangle \, , \end{align} we see that the l.h.s. converges to \(4 p_\star^2\), and the sum of the first two terms on the r.h.s. to \(2 p_\star^2\), which implies \(\langle g_n - f , g_m - f \rangle \to p_\star^2\). We therefore have a Cauchy sequence: \begin{align} \| g_n - g_m \pm f \|^2 = \| g_n - f \|^2 + \| g_m - f \|^2 - 2 \langle g_n - f , g_m - f \rangle \to 2 p_\star^2 - 2 p_\star^2 = 0 \, . \end{align}
(b)
Since \(\mathbb{G}\) is a closed subset of a complete space, it is itself complete. Hence \(( g_n )_n\) is a Cauchy sequence on a complete space which means there exists \(\hat{g} \in \mathbb{G}\) such that \(g_n \to \hat{g}\) with \(\| \hat{g} - f \| = p_\star\).
(c)
Let \(\tilde{g} \in \mathbb{G}\) be element \(\tilde{g} = \hat{g}\) with \(\| \tilde{g} - f \| = p_\star\). We again have \(\| \tfrac{\hat{g} + \tilde{g}}{2} - f \| \leq \tfrac{1}{2} \| \hat{g} - f \| + \tfrac{1}{2} \| \tilde{g} - f \| = p_\star\). Hence the uniqueness follows by an argument analogous to that in (a).
(d)
No. For example, projection on the open unit ball does not result in a well-defined \(\hat{g}\) minimiser.