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[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.]
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