# Maximums and Supremums - vectors

1. Feb 14, 2012

### Ted123

1. The problem statement, all variables and given/known data

Let $${\bf u}(x) = \begin{bmatrix} u_1(x) \\ u_2(x) \end{bmatrix}$$ for $x\in [1,2]$.

$\displaystyle \sup_{\substack{x\in [1,2] }} \| {\bf u} \|$ where $\| {\bf u} \| = \text{max}_j |u_j|$ and am given that $|u_1(x)| \leqslant 5$ and $|u_2(x)| \leqslant 10$.

3. The attempt at a solution

If I was given that $|u_1(x)| = 5$ and $|u_2(x)| = 10$ I'd know how to do it:

$\| {\bf u} \| = \text{max}(|u_1(x)| , |u_2(x)|) = \text{max}(5,10) = 10$

but how do I use the bounds? Do I say:

$\| {\bf u} \| = \text{max}(|u_1(x)| , |u_2(x)|) \leqslant \text{max}(5,10) = 10$

and therefore $\displaystyle \sup_{\substack{x\in [1,2] }} \| {\bf u} \| = 10$ ?