# Finding a Sequence in L^1

1. Oct 28, 2011

### slamminsammya

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?