On the multiplicative completion of certain basic sequences in $L^{p},$ $1<p<\infty$

Abstract:

Boas and Pollard proved that given any basis $\{ {f_n}\} _{n = 1}^\infty$ for ${L^2}(E)$ one can delete the first k basis elements and then find a bounded measurable function M such that $\{ M{f_n}\} _{n = k + 1}^\infty$ is total in ${L^2}(E)$, that is, the closure of the linear span of the set $\{ M{f_n}:n \geq k + 1\}$ is ${L^2}(E)$. We improve this result by weakening the hypothesis to accept bases of ${L^p}(E),1 < p < \infty$, and strengthening the conclusion to read serially total, that is, given any $f \in {L^2}(E)$ one can find a sequence of reals $\{ {a_n}\} _{n = k + 1}^\infty$ such that $\Sigma _{n = k + 1}^\infty {a_n}M{f_n}$ converges to f in the norm. We also show that certain infinite deletions are possible.