Let · be a norm on ℝ^n. Prove that there is a constant C0 such that the entries of every 𝐯=(v1,

Question

Let · be a norm on ℝ^n. Prove that there is a constant C>0 such that the entries of every 𝐯=(v1,

Accepted Answer

Step 1: Understand the problem statement. We need to show that there exists a constant u005C( C u003E 0

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Prove that there is a constant $C>0$ such that the entries of every $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)^T \in \mathbb{R}^n$ are all bounded, in absolute value, by $\left|v_i\right| \leq C\|\mathbf{v}\|$.