my analysis book claims that for $u \in U \subset E$, $E$ is a normed space of finite dimension, then the claims $u \to 0$ and $r(u)/||u|| \to 0$ are independent of the norm (I suppose this means that if this holds for an specific norm, then it holds for every norm on $E$). I know that every two norms on $E$ are equivalent, but how does that imply that the convergences are the same? the first one is clear to me (since we're taking limits in a hausdorff space without changing the topology)
context: the book is claiming that the notion of differentiability at a point holds beyond $\mathbb{R}^n$ and $\mathbb{R}^m$. it says that we can change that by any finitie-dimensional, normed vector spaces $E$ and $F$ in such a way that $f:E \to F$ is differentiable at $u\in E$ if there's an open set $U\ni u$, a linear mapping $T:E \to F$ and a continuous function $r: E \to F$ s.t. $f(u+h) = f(u) + T(h) + r(h)$ with $r(h)/||u|| \to 0$ as $u \to 0$
@Thorgott equivalence of norms? like $\alpha||u||_1 \le ||u||_2 \le \beta ||u||_1$?
