Question

Let $\|\cdot\|_2$ denote the usual Euclidean norm on $\mathbb{R}^n$. Determine the constants in the norm equivalence inequalities $c^{\star}\|\mathbf{v}\| \leq\|\mathbf{v}\|_2 \leq C^{\star}\|\mathbf{v}\|$ for the following norms: (a) the weighted norm $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$, (b) the norm $\|\mathbf{v}\|=\max \left\{\left|v_1+v_2\right|,\left|v_1-v_2\right|\right\}$.

   Let $\|\cdot\|_2$ denote the usual Euclidean norm on $\mathbb{R}^n$. Determine the constants in the norm equivalence inequalities $c^{\star}\|\mathbf{v}\| \leq\|\mathbf{v}\|_2 \leq C^{\star}\|\mathbf{v}\|$ for the following norms: (a) the weighted norm $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$,
(b) the norm $\|\mathbf{v}\|=\max \left\{\left|v_1+v_2\right|,\left|v_1-v_2\right|\right\}$.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 3, Problem 37

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Step 1

### Part (a): Weighted norm $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$ ** Show more…

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Let $\|\cdot\|_2$ denote the usual Euclidean norm on $\mathbb{R}^n$. Determine the constants in the norm equivalence inequalities $c^{\star}\|\mathbf{v}\| \leq\|\mathbf{v}\|_2 \leq C^{\star}\|\mathbf{v}\|$ for the following norms: (a) the weighted norm $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$, (b) the norm $\|\mathbf{v}\|=\max \left\{\left|v_1+v_2\right|,\left|v_1-v_2\right|\right\}$.

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