In particular, are there any countable bases of $L^1([0,1])$?
2026-05-14 19:03:54.1778785434
Give examples of a basis of $L^1([0,1])$
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Yes, take for example the Haar basis consisting of functions of the form $h_I=\chi_{I_l}-\chi_{I_r}$ where $I$ is a standard dyadic interval and $I_l$, $I_r$ its left, respectively right child intervals and $\chi_I$ are (depending on your needs, properly normalized) characteristic functions.
This constitues a countable (Schauder) basis for $L^p([0,1])$ when $p\ge 1$ and $p<\infty$.