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Homework Help: Determining whether these functions are norms

  1. Nov 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine whether these functions are norms:

    a. ||.||2/3: ℝn → ℝ
    ||(x1,...,xn)||2/3 = (Ʃ|xi|2/3)3/2.

    b. If ||.|| is a norm and λ [itex]\in[/itex] ℝ\{0}
    ||x||λ = λ||x||.

    c. If ||.||1 and ||.||2 are norms:
    ||x||max = max{||x||1,||x||2}.

    d. If ||.||1 and ||.||2 are norms:
    ||x||min = min{||x||1,||x||2}.

    2. Relevant equations

    A norm on a vector space X is a real-valued function on X whose value at an x [itex]\in[/itex] X is denoted by ||x|| and which has the properties:

    1. ||x|| ≥ 0
    2. ||x|| = 0 ⇔ x = 0
    3. ||αx|| = |α| ||x||
    4. ||x + y|| ≤ ||x|| + ||y||
    here x and y are arbitrary vectors in X and α is any scalar.

    3. The attempt at a solution

    What I would do is check if the functions have the four properties listed above.

    I think that ||.||λ is not a norm because λ||x|| can be negative.

    I know that ||.||2/3 has the first three properties, but I don't know how to check if it has the fourth one.

    And as for ||.||max and ||.||min, I don't even know what max{} and min{} mean.

    Any help would be appreciated! Thanks.
  2. jcsd
  3. Nov 17, 2012 #2

    That's right. Is that the only thing that goes wrong? What if we stipulate that λ must be positive (instead of just nonzero)?

    What are the easiest vectors to check? The second easiest?

    If you had to guess, what would say that max{} and min{} meant? Seriously, take a guess. I can almost guarantee you that you'll be right (assuming you're a native speaker of English).
  4. Nov 18, 2012 #3
    #1 ||x||λ ≥ 0 ⇔ λ > 0 (λ [itex]\in[/itex] ℝ\{0})

    #2 ||x||λ = 0 ⇔ x = 0 (since λ ≠ 0 and ||x||λ = λ||x||)

    #3 ||αx||λ = λ ||αx|| = λ |α| ||x|| = |α| ||x||λ

    #4 ||x+y||λ ≤ ||x||λ + ||y||λ

    λ ||x+y|| ≤ λ (||x|| + ||y||)

    and ||x+y|| ≤ (||x|| + ||y||) holds because ||.|| is a norm.

    Thus, ||.||λ must be a norm for λ > 0. Am I right?
    I don't know. Can you tell me? :)

    For ||.||2/3 #1 and #2 are trivial.

    #3 ||αx||2/3 = (Ʃ|αxi|2/3)3/2 = (Ʃ|α|2/3|xi|2/3)3/2 = (|α|2/3 Ʃ |xi|2/3)3/2 = |α| (Ʃ |xi|2/3)3/2 = |α| ||x||2/3

    I don't know how to check #4:

    (Ʃ |xi+yi|2/3)3/2 ≤ (Ʃ |xi|2/3)3/2 + (Ʃ |yi|2/3)3/2
    Well, even though English is not my native language I'm quite sure max and min stand for maximum and minimum respectively. I'm just not very familiar with them as mathematical symbols. However, if they mean what I think they mean, I suppose that #1 and #2 are trivial for ||.||max and ||.||min.

    I'm not sure about #3, but I think that

    ||αx||max = max{||αx||1,||αx||2} = max{|α| ||x||1,|α| ||x||2} = |α| max{||x||1,||x||2} = |α| ||x||max

    and the same for ||.||min.

    I don't know how to check #4 for these functions either :(
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