Lee writes on page 1 of his Smooth Manifolds book,
This book is about smooth manifolds. In the simplest terms, these are spaces that locally look like some Euclidean space $\mathbb{R}^n$, and on which one can do calculus
Wikipedia (and Lee) defines Euclidean space as an inner product space with the norm derived from that inner product as $\|x\|={\sqrt {x\cdot x}}$
Which parts of this Euclidean space are really necessary for defining and working with smooth manifolds?
At the very least we need a norm to define derivatives and smoothness, but do we really need the Euclidean norm as opposed to any norm?
Also, do we require that there is an inner product associated with $\mathbb{R}^n$ and that the norm is derived from the inner product?
Note, I am only interested in the answer in the context of smooth manifolds (as opposed to considering this question more broadly when defining Riemmanian Geometry).