# How to proof that these two statements are equivalent?

Discussion in 'Analysis and Topology' started by Polleei, May 17, 2022.

1. ### Polleei

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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.

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

Polleei, May 17, 2022

2. ### MathLover1

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d(x, y) = ||x−y|| for all x, y ∈ W.

Proof:
Given a norm, the function d(x, y) = ||x − y|| is translation in-variant in the sense that d(x + a, y + a) = ||x + a − y − a|| = d(x, y), scalar homogeneous in the sense that d(λx, λy) = |λ|d(x, y).

Conversely if d(x, y) is translation invariant, then d(x, y) = d(x − y, y − y) = d(x − y, 0) and we
can define ||x|| = d(x, 0) so that d(x, y) = ||x − y||.

We will now show that the axioms for the norm correspond precisely to the axioms for a distance.

d satisfies the triangle inequality <=> the norm ||.|| does

d(x, y) ≤ d(x, z) + d(z, y)

corresponds to ||x − y|| ≤ ||x − z|| +||z − y||.

Similarly d(x, y) = d(y, x) corresponds to ||x − y|| = ||y − x||;
d(x, y) ≥ 0 is ||x − y|| ≥ 0,

while d(x, y) =0 <=> x = y becomes ||x − y|| = 0 <=> x − y = 0.
The scale-homogeneity of the metric supplies the final axiom for the norm. This invariance under translations and scaling has the following easy con-sequences.

MathLover1, May 17, 2022
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3. ### Polleei

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I think i get the idea of the proof, but whats with the image in row 8 and 13? Its just a cross. Is it important?
Otherwise, thank you very much for helping me out Polleei, May 17, 2022
4. ### MathLover1

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should be: <=>

MathLover1, May 17, 2022
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5. ### Polleei

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You dropped this :crown:

Polleei, May 17, 2022
6. ### MathLover1

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You dropped this :crown: ??? what do you mean by that

MathLover1, May 17, 2022
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7. ### Polleei

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Means thanks for the help king, thats why a crown Polleei, May 17, 2022
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8. ### MathLover1

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then should be queen 