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Calculating a metric from a norm and inner product.

  1. Feb 12, 2012 #1
    I typed the problem in latex and will add comments below each image.

    norm1.png

    The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I say?

    norm2.png

    For this one, I'm not sure if I did it correctly. So could someone just verify that my solution is correct?

    norm3.png

    I'm really not sure how to do this one. I'm trying to relate it to the unit circle from trigonometry, but that had an x and y coordinate...For this one, would I just choose sample points (1, 0), (0, 1), (-1, 0), and (0, -1) and sketch the norm at each of these? But the norm only takes 1 value, not two...
     
  2. jcsd
  3. Feb 12, 2012 #2
    Q1. C([0,1])="the function space of continuous functions defined on the interval [0,1]" so [0,1] is fixed, you cannot shift it around.

    Q2. last step is wrong.

    Q3. a Ball={set of all x in ??? such that Norm(x-0)<=radius"} it is implied in the question what ??? is.
     
  4. Feb 12, 2012 #3
    thank you, i have the problem complete now!
     
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