  Can polynomial interpolation be extended to a Banach space setting? Are
tensors whose elements are non-commutative Banach space elements legitimate
objects with notable analytic and algebraic properties? Here we explore these
questions and present the ground work conducted to successfully interpolate a
left-sided polynomial whose inputs are general Banach space elements, when the
Banach space admits an associative algebra, submultiplicative norm and a
multiplicative identity. We start with introductions and propose theorems and
lemmas equipped for general tensors over a Banach space for those whose
research interests lie outside of Lagrange interpolation, transitioning into
algebraic properties inspired by Maksimov's work on cubic matrices, gradually
narrowing down to the interpolation itself for left-sided polynomials in the
last section of the article.
