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Math Help - Norms and Cauchy Sequences

  1. #1
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    Norms and Cauchy Sequences

    Hi!

    Two norms  ||\bullet || and  |||\bullet ||| on a vector space V are called equivalent if there exist 0 < a <= b such that

     a ||\bullet || \le |||\bullet ||| \le b ||\bullet ||


    -------

     ||\bullet || and  |||\bullet ||| are equivalent
    Show that

    A sequence is a cauchy-sequence on  ||\bullet || <=> A sequence is a cauchy-sequence on  |||\bullet |||


    I do not have any clue. Any help would be much appreciated.

    best regards
    Rapha
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  2. #2
    Moo
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    Hello,

    If (a_n)_{n \in \mathbb{N}} is a Cauchy sequence with the \|.\| norm, then write the definition :

    \forall \varepsilon' >0,~ \exists N \in \mathbb{N},~ \text{such that } \forall m,n > N,~ \|x_m-x_n\|< \varepsilon'

    But we know that there exists b such that :
    b \|.\| \geqslant |||.|||
    that is \|.\| \geqslant \frac 1b \cdot |||.||| (which is possible since b>0)

    So going back to the definition, we have :
     \frac 1b |||x_m-x_n||| \leqslant \|x_m-x_n\|< \varepsilon'

    Hence :
    |||x_m-x_n|||< b \varepsilon'

    Let \varepsilon=b \varepsilon' and it'll give the Cauchy sequence definition for the |||.||| norm.
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  3. #3
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    Hey Moo,

    thank you so much, you are awesome!
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