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Math Help - Metric spaces

  1. #1
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    Metric spaces

    Let

    X:=\{{(a_n)^\infty_{n=0} : \sum^{\infty}_{n=0}|a_n|<\infty}\}

    be the space of absolutely covergent sequences.

    Define the l^1 and l^{\infty} metric on this space by

    d_{l^1}((a_n)^\infty_{n=0}, (b_n)^\infty_{n=0}):=\sum^{\infty}_{n=0}|a_n-b_n|

    d_{l^\infty}((a_n)^\infty_{n=0}, (b_n)^\infty_{n=0}) := sup_{n\in\mathbb{N}}|a_n-b_n|

    First, I want to show that there exist sequences x^{(1)}, x^{(2)}, ... of elements of X which are convergent with respect to the d_{l^{\infty}} metric but not with respect to the d_{l^1} metric.

    Second, that any sequence which converges in the d_{l^1} metric, automatically converges in the d_{l^{\infty}} metric.

    Thanks
    Last edited by bram kierkels; September 4th 2010 at 10:58 AM.
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  2. #2
    Junior Member
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    A series that converges in the sup-metric, but not in the 1-metric is for example this:
    x_0=(1,0,0,...)
    x_1=(1/2,1/2,0,0,...)
    x_2=(1/3,1/3,1/3,0,0,...)
    x_3=(1/4,1/4,1/4,1/4,0,0,...)
    ...
    x_n=(1/n,1/n,...,1/n,0,0,...) (n times 1/n)
    ...
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