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Maximums and Supremums - vectors

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data

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

    I'm asked to find:

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

    3. The attempt at a solution

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

    [itex]\| {\bf u} \| = \text{max}(|u_1(x)| , |u_2(x)|) = \text{max}(5,10) = 10[/itex]

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

    [itex]\| {\bf u} \| = \text{max}(|u_1(x)| , |u_2(x)|) \leqslant \text{max}(5,10) = 10[/itex]

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