Measure, Integration & Real Analysis · Problem 6B.2

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · complex-numbers-absolute-value-bounds

Problem

Suppose

z \in \mathbf{C}

. Prove that

\max\{|\operatorname{Re} z|, |\operatorname{Im} z|\} \leq |z| \leq \sqrt{2}\, \max\{|\operatorname{Re} z|, |\operatorname{Im} z|\}.