Exercise10.5

Show that the following inequalities hold for all vectors \(x \in \Cm\text{.}\) If possible, give a nontrivial example for which equality holds.

  1. \(\| x \|_\infty \leq \| x\|_2\)
  2. \(\|x \|_2 \leq \sqrt{m}\| x\|_\infty\)
  3. \(\| x \|_2 \leq \| \sqrt{x\|_1 \| x\|_\infty}\)
  4. \(\|x\|_\infty \leq \|x\|_2 \leq \|x\|_1\)

Which, if any, of these inequalities extend to the matrix norms induced by these vector norms?

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