I have already asked here a few questions about polynomials in Banach spaces (Counterexample of polynomials in infinite dimensional Banach spaces, Mujica's "Complex analysis in Banach spaces" exercise 1.2.B, ...).
I am doing my bachelor thesis in that topic and I thought that it would be nice to give some kind of historical overview about the origin of this kind of polynomials. Specially because the definition didn't seem very natural to me at first and when I present the thesis there will be people not familiar with the concept.
I've been looking for some kind of reference where I could see which motivated the extension of the definition of a polynomial to Banach spaces and where it first appeared but I couldn't find any.
Is there any good reference where I can learn a bit in the subject?
Here is the definition of polynomial between Banach spaces:
A map $P$ is an m-homogeneous polynomial from $E$ to $F$ if there is a m-linear map $A$ from $E^m$ to $F$ such that $P(x)=A(x, \dots, x)$.
$P$ is a polynomial of degree at most $m$ if $P = P_0 + \dots + P_m$ where each $P_j$ is an j-homogeneous polynomial.