Commensurable and the stronger one?

[jaavier1]

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




[tex]
\ 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...

 

[Aaron Barnard]

The metrics $d_\infty$, $d_2$, and $d_3$ are commensurable since they all measure the same thing: the distance between two functions in the space $C^1 \left[ {0,T} \right]$. Of the three, $d_\infty$ is the strongest metric as it measures the maximum distance between two points, while the other two metrics measure the average distance. The metric $d$ is not commensurable with the other three because it measures both the maximum distance and the rate of change (i.e., the derivative) between two points.