It is well-known that $\mathcal{C}[0,1]$, the space consisting of all scalar-valued continuous functions over the unit interval, has a monotone Schauder basis. In fact, we can construct such a basis from a countable dense subset, say $D$, of $[0,1]$, and considering the projections that, given $F\subseteq D$ finite, map a function $f\in \mathcal{C}[0,1]$ into the piecewise linear function with nodes in $(x,f(x))$, $x\in F$.
Unfortunately, this canonical basis is not bimonotone. Is there such a basis for $\mathcal{C}[0,1]$? Note that a positive answer would imply that any weakly null sequence in any Banach space would have an approximately bimonotone subsequence.
