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Math Help - finding a sequence of continuous bounded functions

  1. #1
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    finding a sequence of continuous bounded functions

    I want to find a sequence (f_k)of continuous bounded functions on [0,1]

    converging pointwise to a continuous limit, but such that

    the sup-norm of f_k tends to infinity when k tends to infinity...

    is there any?
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  2. #2
    Senior Member jakncoke's Avatar
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    Nope, i don't think there are any. Since if  f_k \rightarrow f pointwise then
     \lVert f_k \lVert_{sup} \rightarrow \lVert f \lVert_{sup} which would mean f is not bounded on [0, 1] (since  \lVert f_k \lVert_{sup} \rightarrow \infty ) thus it must not continuous on [0, 1].
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  3. #3
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    Quote Originally Posted by losin90 View Post
    I want to find a sequence (f_k)of continuous bounded functions on [0,1]

    converging pointwise to a continuous limit, but such that

    the sup-norm of f_k tends to infinity when k tends to infinity...

    is there any?
    What you need to do is to ensure that the function f_k has a narrow spike somewhere. For example, define f_k(x) = k^2x in the interval 0\leqslant x<1/k, f_k(x) = \tfrac2k - k^2x in the interval 1/k\leqslant x<2/k, and f_k(x) = 0 in the remainder of the unit interval. Then f_k(x)\to0 as k\to\infty, for each (fixed) point x in the unit interval.

    Quote Originally Posted by jakncoke View Post
    Nope, i don't think there are any. Since if  f_k \rightarrow f pointwise then
     \lVert f_k \lVert_{sup} \rightarrow \lVert f \lVert_{sup} which would mean f is not bounded on [0, 1] (since  \lVert f_k \lVert_{sup} \rightarrow \infty ) thus it must not continuous on [0, 1].
    That is not true, because pointwise convergence does not imply convergence in the sup norm.
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