Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector.

Question

[Dual variational problems] Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector. Note that \(Q=I-P\), where \(P\) in the orthogonal projection into \(V\)
a) Show that \(\max _{\{z \perp V,\|z\|=1\}}\langle x, z\rangle=\|Q x\|\).
b) Show that \(\min _{v \in V}\|x-v\|=\|Q x\|\).
[Remark: dual variational problems are a pair of maximum and minimum problems whose extremal values are equal.]