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Vector spaces

  1. Dec 13, 2004 #1
    hi

    this problem requires calculus, as it also concerns Vector spaces, i solved alot of Vector spaces problems, either the subset is matrix or ordered pairs.

    this question says:
    Which of the following subsets of the vector space C(-inf, inf) are subspaces:

    (note: C(-infinity, infinity) is vector of functions defined for all real numbers)

    - All integrable functions.
    - All bounded functions.
    - All functions that are integrable on [a,b].
    - All functions that are bounded on [a,b].

    i just need detailed explanantion if possible...thnx for ur efforts
    ur efforts will be appreciated
    thank u
     
  2. jcsd
  3. Dec 13, 2004 #2
    Just check the list of things a subspace must fulfil... For example, if f and g are integrable, is f + g? Is tf integrable for all real t? Etc.
     
  4. Dec 14, 2004 #3
    thnxxxx alot Muzza for ur help

    wish u good luck
    good bye
     
  5. Dec 14, 2004 #4

    arildno

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    Okay, let's take the second one:
    Suppose f and g are bounded.
    This means, there exist a number "A" so that |f|<=A for all x.
    There exist "B" so that |g|<=B for all x.

    But, by the triangle inequality, we have:
    |f+g|<=|f|+|g|<=A+B for all x
    Hence, f+g is bounded as well.
    Can you finish that proof?

    Hope this has given you some ideas..

    EDIT:
    Corrected an equality sign to an inequality sign .
     
    Last edited: Dec 14, 2004
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