Show that if $\lVert \cdot \rVert$ is a Norm on the Scalar Field $\mathbb{F}$, then there exists a positive number $\lambda > 0$ such that $\lVert x \rVert = \lambda |x|$ for all $x \in \mathbb{F}$
We know that both $0 \leq \lVert x \rVert < \infty$ and $ 0 \leq |x| < \infty$. So intuitively there must be a positive scalar mapping $\lVert x \rVert$ to $|x|$. However, I don't know how to go about proving this rigorously, in particular proving there is one $\lambda > 0$ such that the inequality holds for all $x$.
Best Answer
Be definition of a norm $\|x\|=\|(x)(1)\|=|x|\|1\|$. So the identity holds with $\lambda =\|1\|$.