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Math Help - Urgent help needed - matric space & continuity

  1. #1
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    Urgent help needed - matric space & continuity

    i HAVE DIFFICULTIES WIH AN ASSIGMENT. ANY HELP ?


    2A

    Let M be a set of non-negative real valued function on the real valued axe with

    M = { f: R→ R | f(x) ≥ 0 for alle x belongs to R }

    Show that the matric distance

    d(f,g) = supp {|e^(-f(x)) -e ^-g(x)|, | x belongs to R}, f,g belongs to R

    Defines a metric d on the set M


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    2B


    Let for n belongs to N, the functions fn and gn belongs to M, n belongs to N be defined by

    fn(x) = x^2 if |x| ≤ n
    n^2 if |x| > n

    and

    gn(x) = x^2 if |x| ≤ n
    = 0 if |x| > n

    Do the functional series fn(x) (n belongs to N)
    and the functional series gn(x) (n belongs to N) converges uniformly on R ?

    If not unifrom convergent for what reasons are the functions series fn and gn not unifrom convergent ?

    For each of the two functions determine its convergence on the metric space (M,d) ?


    ---

    BR
    mathcph

    mathstudentdk@yahoo.com
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  2. #2
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    Quote Originally Posted by mathcph View Post
    2A

    Let M be a set of non-negative real valued functions on the real axis with

    M = { f: R→ R | f(x) ≥ 0 for all x in R }

    Show that the metric distance

    d(f,g) = \sup \{|e^{-f(x)} -e ^{-g(x)}|:\; x \in \mathbb{R}\},\quad f,\,g \in M

    defines a metric d on the set M.
    First, is d well defined (in other words, is that sup finite)? —Yes, because the negative exponentials never take a value greater than 1.

    Next, are the axioms satisfied?

    (1) Is d(f,g)≥0? —Yes, obviously.
    (2) If d(f,g) = 0, does that imply that f = g? (I'll leave that one to you.)
    (3) Is d(g,f) = d(f,g)? —Yes, obviously.
    (4) Is d(f,h) ≤ d(f,g) + d(g,h)? (Use the ordinary triangle inequality for real numbers here.)

    Quote Originally Posted by mathcph View Post
    2B


    For n in N, let the functions f_n and g_n in M be defined by

    f_n(x) = \begin{cases}x^2 &\text{if }|x| \leqslant n \\<br />
n^2 &\text{if } |x| > n\end{cases}

    and

    g_n(x) = \begin{cases}x^2 &\text{if }|x| \leqslant n \\<br />
  0&\text{if } |x| > n\end{cases}

    Do the functional series f_n(x) (n in N)
    and the functional series g_n(x) (n in N) converge uniformly on R?

    If not uniformly convergent for what reasons are the function series f_n and g_n not uniformly convergent ?

    For each of the two functions determine its convergence in the metric space (M,d).
    To see that neither of these series of functions converge, notice that in both cases the pointwise limit of the series is the function f(x)=x^2. Then ask yourself whether the difference between the n'th function in the series and the limit function can be made small for all x in R by taking n large enough.

    On the other hand, d(f_n,f) and d(g_n,f) both tend to 0 as n→∞.
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  3. #3
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    Urgent help needed - matric space & continuity - follow up

    Can you be more specific about the pointwise and not the uniform convergence of each of the two functions on the metric space ?

    I am no further than I was when I raised the questions. What you writte is wellknown stoff to me. The next step ? I do not know.

    mathcph
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