Subsystems of the Schauder system that are quasibases for $L^p[0,1], 1 \le p < +\infty$

Abstract:

We show that if $\Phi =\{\varphi _i\colon i=1,2,\ldots \}$ is a subsystem of the Faber-Schauder system, and if $\Phi$ is complete in $L^2[0,1]$, then $\Phi$ is a quasibasis for each space $L^p[0,1]$, $1\le p<+\infty$. Although it follows from the work of Ul’yanov that each element of $L^p[0,1]$ can be represented by a Schauder series that converges unconditionally to the function, in the metric of the space, it proves to be the case that none of the aforementioned systems is an unconditional quasibasis for any of the $L^p$-spaces herein considered.