91Ó°ÊÓ

Let \(\|\cdot\|\) denote any norm on \(\mathbb{C}^{m}\). The corresponding dual norm \(\|\cdot\|^{\prime}\) is defined by the formula \(\|x\|^{\prime}=\sup _{\|y\|=1}\left|y^{*} x\right|\) (a) Prove that \(\|\cdot\|^{\prime}\) is a norm. (b) Let \(x, y \in \mathbb{C}^{m}\) with \(\|x\|=\|y\|=1\) be given. Show that there exists a rank-one matrix \(B=y z^{*}\) such that \(B x=y\) and \(\|B\|=1\), where \(\|B\|\) is the matrix norm of \(B\) induced by the vector norm \(\|\cdot\| .\) You may use the following lemma, without proof: given \(x \in \mathbb{C}^{m}\), there exists a nonzero \(z \in \mathbb{C}^{m}\) such that \(\left|z^{*} x\right|=\|z\|^{\prime}\|x\|\)