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Hey there, i'm having a lot of trouble with this task right here. Let W be a real vector space and d a metric on it. Proof that the two statements are equivalent: (1) The metric d is both homogeneous and translation invariant and (2) There is exactly one norm ∥ · ∥ on W such that the metric induced by this norm with d agrees, i.e. d(x, y) := ∥x−y∥ for all x, y ∈ W.
What i found out already:
The metric d is homogeneous if d(λx,λy) = |λ|d(x, y) for all x, y ∈ W and all λ ∈ R
Also, we call the metric d translation-invariant, if for all x, y,z ∈ W we have d(x + z, y + z) = d(x, y)
I already spoke to many of my classmates about this but noone seemed to have a clue on how to proof this. I'd be very thankfull for support on this task
Best regards
What i found out already:
The metric d is homogeneous if d(λx,λy) = |λ|d(x, y) for all x, y ∈ W and all λ ∈ R
Also, we call the metric d translation-invariant, if for all x, y,z ∈ W we have d(x + z, y + z) = d(x, y)
I already spoke to many of my classmates about this but noone seemed to have a clue on how to proof this. I'd be very thankfull for support on this task

Best regards