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Am I understanding "supremum" correctly

  1. Jan 26, 2016 #1
    If let's say I have an expression:
    ##x\leq y##

    Since the supremum is defined as the "least upper bound," does this make sup(x) for this case ##x=y## or is it ##x = \infty##?
     
  2. jcsd
  3. Jan 26, 2016 #2

    The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.
     
  4. Jan 26, 2016 #3
    In this case I am asking for the supremum of x, sup(x), which would be the least upper bound of x. Would it be x = y?
     
  5. Jan 26, 2016 #4

    PAllen

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    Maybe fixed after you commented, but OP reads sup(x). Then, sup(x)=y is correct for the set of x ≤ y, assuming y is a given value. Of course, this is a totally trivial case.
     
  6. Jan 26, 2016 #5

    PAllen

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    You wouldn't say x=y, you would say sup(x)=y, for the specified set.
     
  7. Jan 26, 2016 #6
    Thanks. If instead we want sup(y), would that be infinity?

    I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.
     
  8. Jan 26, 2016 #7
    ah yes, thanks
     
  9. Jan 26, 2016 #8

    WWGD

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    But as you wrote it, it seems y is just one number, and not a set, so, trivially, sup{y}=y (on most "reasonable" choices of orderings , where ## a \leq a ##)
     
  10. Jan 26, 2016 #9

    PAllen

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    That depends on how you define the set against which x≤y is defined. That is, given x, and the set of y such that x ≤ y of real numbers, then sup(y) does not exist. If, instead, you use the set of extended reals that includes +/- ∞ as wellas the conventional reals, then, indeed, sup(y)=∞.
     
  11. Jan 26, 2016 #10



    I see. This was not a very good hypothetical example. This is a question in direct relation with another thread of mine, https://www.physicsforums.com/threads/is-this-proof-valid-of-an-infty-norm-valid.854189/
    where I am considering the infinite matrix norm, who's definition involves a supremum over a set of numbers.
     
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