Weaving Schauder frames

We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame if and only if there is a uniform constant $C\geq 1$ such that every weaving is a $C$-approximate Schauder frame. We also study weaving Schauder bases, where it is necessary to introduce two notions of weaving. On one hand, we can ask if two Schauder bases are woven when considered as Schauder frames with their biorthogonal functionals, and alternatively, we can ask if each weaving of two Schauder bases remains a Schauder basis. We will prove that these two notions coincide when all weavings are unconditional, but otherwise they can be different. Lastly, we prove two perturbation theorems for approximate Schauder frames.