How to proof that these two statements are equivalent?

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

  1. Polleei

    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.

    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
     
    Polleei, May 17, 2022
    #1
  2. Polleei

    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

    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 :D
     
    Polleei, May 17, 2022
    #3
  4. Polleei

    MathLover1

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    should be: <=>
     
    MathLover1, May 17, 2022
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  5. Polleei

    Polleei

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    You dropped this :crown:
     
    Polleei, May 17, 2022
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  6. Polleei

    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

    Polleei

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

    MathLover1

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    then should be queen :)
     
    MathLover1, May 17, 2022
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