Exercise 3.6

Let $\text{∥⋅∥}$ denote any norm on ${ℂ}^m$. The corresponding dual norm $\parallel \cdot {\text{∥}}^{\prime }$ is defined by the formula

(a) Prove that $\parallel \cdot {\text{∥}}^{\prime }$ is a norm.
(b) Let $x,y\in {ℂ}^m$ with $\parallel x\parallel =\parallel y\parallel =1$ be given. Show that there exists a rank-one matrix $B=\mathit{yz}\ast$ such that $\mathit{Bx}=y$ and $\parallel B\parallel =1$, where $\parallel B\parallel$ is the matrix norm of $B$ induced by the vector norm $\parallel \cdot {\text{∥}}^{\prime }$ You may use the following lemma, without proof: given $X\in {ℂ}^m$, there exists a nonzero $z\in {ℂ}^m$ such that $|z\ast x|=\parallel z{\text{∥}}^{\prime }\parallel x\parallel$.