slamminsammya
Oct28-11, 04:29 PM
Let L^1(\mathbb{R}^d) be the space of Lebesgue integrable functions in d-dimensional real space. Find a sequence of functions f_n such that ||f_n||_{L^1}=1 for each n and yet ||f_n-f_m||_{L^1}=1.
One way to do this problem is to explicitly construct such functions. Given the sets E_n=[0,1/2]\cup [n,n+1/2], the functions \chi_{E_n} 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 L^1 is infinite dimensional, we should be able to find an f_2 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 B_1((0))\cap B_1(f_1)\cap \dots \cap B_1(f_n) 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?
One way to do this problem is to explicitly construct such functions. Given the sets E_n=[0,1/2]\cup [n,n+1/2], the functions \chi_{E_n} 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 L^1 is infinite dimensional, we should be able to find an f_2 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 B_1((0))\cap B_1(f_1)\cap \dots \cap B_1(f_n) 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?