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A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define a norm, and vice versa. So there must exist a non-translation invariant but homogeneous metric linear space. However, I have tried my best to think about it, but gained nothing. Please help me.

The related interpretation can be found here.

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Make $\mathbf{R}^2$ a metric space using the so-called "French railway distance." The distance $d(A,B)$ from a point $A$ to $B$ is the usual distance $AB$ if they lie on the same ray from the origin $O$. Otherwise, $d(A,B) = AO + OB$.

Then this is a distance function, and it is homogeneous in the sense that $d(ax,ay) = |a|d(x,y)$. However, it is not translation invariant.

The name refers to the idea that to travel between any two points in France, you must pass through Paris, unless the points happen to be on the same railway line to Paris. (This interpretation makes it obvious that $d$ is a distance.)

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  • $\begingroup$ for this metric, the regarding linear operation is continuous regarding to the metric? $\endgroup$ – David Chan Jan 10 '14 at 1:28

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