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Commensurable and the stronger one?

  1. Oct 23, 2008 #1
    i have the next metrics on the $C^1 \left[ {0,T} \right]$ space

    \ d_\infty (x,y) = \sup {\rm{ metric}} \\
    d_2 (x,y) = \left\{ {\int_0^T {\left[ {x(t) - y(t)} \right]^2 dt} } \right\}^{1/2} \\
    d_3 (x,y) = \left\{ {\int_0^T {\left| {x(t) - y(t)} \right|^3 dt} } \right\}^{1/3} \\
    d(x,y) = d_\infty (x,y) + d_\infty (\dot x,\dot y),{\rm{ where }}\dot x = \frac{{dx}}{{dt}}

    each one generate a topology in that space (\[T_\infty\], \[T_\2\], \[T_\2\],\[T_\0\] )

    the questions are, which of them are commensurables?
    if there are 2 commensurables which is the stronger one?

    Note that the metrics are equivalet...
  2. jcsd
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