Let $x_1<\dots<x_n$, then are the functions $$ 1_{[0,x_i]}, $$ linearly independent in $L^1$?
It seems obvious to me but I can't show it...
2026-03-29 10:46:41.1774781201
Linear independence of Indicators
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Suppose $\sum c_iI_{[0,x_i]}=0$. Then $\sum c_iI_{[0,x_i]}(x)=0$ for almost all $x$. Take $x \in (x_{n-1},x_n)$ for which the equation holds to get $c_n=0$. Then take $x \in (x_{n-2},x_{n-1})$ and so on.