Results 1 to 3 of 3

Math Help - abstract algebra

  1. #1
    Junior Member
    Joined
    Feb 2014
    From
    USA
    Posts
    33
    Thanks
    3

    abstract algebra

    given the function f: X --> R+ that is bounded above, proove that sup(f^2)=sup^2(f)

    I did the proof when given two functions f and g, we have that sup(f*g) is less then or equal to sup(f) * sup(g) using the same hypotheses.
    How may I proove the equality when I have one function only?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,922
    Thanks
    1762
    Awards
    1

    Re: abstract algebra

    Quote Originally Posted by Yeison View Post
    given the function f: X --> R+ that is bounded above, proove that sup(f^2)=sup^2(f)
    I did the proof when given two functions f and g, we have that sup(f*g) is less then or equal to sup(f) * sup(g) using the same hypotheses.
    How may I proove the equality when I have one function only?
    From that, you know that $\sup(f^2)\le\sup^2(f)$. Right?

    What if $\sup(f^2) < \sup^2(f)~?$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: abstract algebra

    Definitions:
    f: set of all values of f
    f2: set of all values of f*f


    Briefly. Let x ϵ f, and note there is a real number a between any two real numbers.


    1) (lubf)2<a<lubf2
    x2<a all x, and a<x2 for some x. contradiction.

    2) lubf2<a<(lubf)2
    x2<a all x and a<x2 for some x. contradiction.

    Therefore: (lubf)2=lubf2

    Details of 1):
    (lubf)2<a:
    let x ϵ f. Then x ≤ lubf and x2 ≤ (lubf)2 all x → x2 < a all x.

    a<lubf2:
    let x ϵ f. Then a < x2 for some x, otherwise a ≥ x2 for all x and a is an upper bound of f2 less than lubf2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 6th 2010, 04:03 PM
  2. abstract algebra
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 4th 2010, 04:54 AM
  3. Replies: 0
    Last Post: April 24th 2010, 12:37 AM
  4. Abstract algebra
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 30th 2010, 11:27 PM
  5. Abstract Algebra
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 24th 2008, 07:32 PM

Search Tags


/mathhelpforum @mathhelpforum