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Finding a Sequence in L^1

  1. Oct 28, 2011 #1
    Let [itex]L^1(\mathbb{R}^d)[/itex] be the space of Lebesgue integrable functions in d-dimensional real space. Find a sequence of functions [itex]f_n[/itex] such that [itex]||f_n||_{L^1}=1[/itex] for each n and yet [itex]||f_n-f_m||_{L^1}=1[/itex].

    One way to do this problem is to explicitly construct such functions. Given the sets [itex]E_n=[0,1/2]\cup [n,n+1/2][/itex], the functions [itex]\chi_{E_n}[/itex] work. But I was wondering if there is a way to do this with linear algebra. My thought was this:

    Clearly we can find a function of norm 1. Then using the fact that [itex]L^1[/itex] is infinite dimensional, we should be able to find an [itex]f_2[/itex] in the intersection of the unit ball around f and the unit sphere. We should then be able to continue in this way, since the intersection [itex]B_1((0))\cap B_1(f_1)\cap \dots \cap B_1(f_n)[/itex] should always contain an element not in our list, owing to the fact that the space is infinite dimensional.

    But I can't quite get the argument right. Is this possible? If so, how do you show it rigorously?
  2. jcsd
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