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Continuity of the the norm

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove the continuity of the norm; ie show that in any n.l.s. N if xn [tex]\rightarrow[/tex] x then [tex]\left|\left|x_n\left|\left|[/tex] [tex]\rightarrow[/tex] [tex]\left|\left|x\left|\left|[/tex]

    3. The attempt at a solution

    i dont know where to start this
    from the definition of convergence xn [tex]\rightarrow[/tex] x as n[tex]\rightarrow[/tex] [tex]\infty[/tex] if [tex]\left|\left|x_n - X\left|\left|[/tex] [tex]\rightarrow[/tex] 0 as n[tex]\rightarrow[/tex] [tex]\infty[/tex]

    do i use this to prove it or am i barking up the wrong tree?
  2. jcsd
  3. Feb 23, 2010 #2


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    Sure you use the definition of convergence. Can you prove the 'reverse triangle inequality'? abs(||a||-||b||)<=||a-b||.
  4. Feb 23, 2010 #3
    let |a| ≥ |b|,
    However, we can write a as a - b + b or a = (a - b) + b

    Then |a| = |a - b + b|. But, |a - b + b| [tex]\leq[/tex] |a - b| + |b| by the Triangle Inequality, and so we have that |a| [tex]\leq[/tex] |a - b| + |b| . Now, subtract |b| from both sides. This gives us: |a|-|b| [tex]\leq[/tex] |a - b|

  5. Feb 23, 2010 #4


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    Sure. That's it. Makes your proof easy, right?
  6. Feb 23, 2010 #5
    thanks a million, i guess i'm just not looking at them right, but now i see
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