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A working example wrt the supremum norm

  1. Dec 20, 2011 #1
    Folks,

    Could anyone give me a working example of a sequence of functions that converges to a function wrt to the supremum norm?

    Thank you.
     
  2. jcsd
  3. Dec 20, 2011 #2

    micromass

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    Depends on what you mean with "working example". A nice example is probably

    [tex]f_n(x)=\sum_{k=0}^n{\frac{x^k}{k!}}.[/tex]

    This converges to [itex]f(x)=e^x[/itex]. This convergence is uniform on each bounded set.
     
  4. Dec 20, 2011 #3
    Would this converge to the same function wrt to the integral norm?
     
  5. Dec 20, 2011 #4

    micromass

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    Yes. It is actually a nice exercise to show that it does. In fact, it suffices in this case to show that

    [tex]\int_a^b|f_n|\rightarrow \int_a^b |f|[/tex]
     
  6. Dec 20, 2011 #5

    LCKurtz

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    Pick a number a with ##0 < a < 1##. Let ##f_n(x) = x^n\hbox{ on }[0,a]##. Then ##\|f_n - 0\|\rightarrow 0##. Note that this fails if ##a=1##.
     
  7. Dec 20, 2011 #6

    micromass

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    LCKurtz' example is a really nice one!! It is nice to notice that it can be generalized into what is known as Dini's theorem:

    If [itex](X,d)[/itex] is a compact metric space and if [itex]f_n:X\rightarrow \mathbb{R}[/itex] are a sequence of functions such that
    - They are monotonically decreasing/increasing
    - They pointswize converge to a function f
    - f is continuous
    then the convergence is uniform.

    It is one of the few cases where pointsiwize convergence implies uniform convergence.

    LCKurtz' example follows with X=[0,a] (with 0<a<1), [itex]f_n(x)=x^n[/itex] and [itex]f(x)=0[/itex]. It is a nice exercise to see what goes wrong in the theorem if a=1.
     
  8. Dec 21, 2011 #7
    Would it be something like this...


    [itex]\displaystyle ||f_n(x)||_1=\int_{a}^{b}| \sum_{n=0}^{\infty} \frac{x^k}{k!}|dx[/itex] for c[a,b]
     
  9. Dec 21, 2011 #8
    Hmmm..but regarding post 3 ie a function f_n the converges to the same function f on both sup and integral norms...this function does not converge to a function but to zero.....?

     
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