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I want to go from general to specific, meaning, if I have the definition of Frechet differentiability for real or complex Banach spaces I want to obtain all the other definitions of differentiability for functions between $\mathbb{R}^m$ and $\mathbb{R}^n$ from that.

Now my question is: How can I get from the Frechet derivative, in the case that $V$ is one-dimensional (so isomorphic to $\mathbb{K}=\mathbb{R,C}$), to the following definition of differentiablity (which, as far as I know, is the definition for differentiablity for curves):

$f:\mathbb{K} \rightarrow W$ is differentiable at $x\in U\subseteq \mathbb{K},\ U$ open if the limit of the function $F:\mathbb{K}\setminus \{0\} \ni h\mapsto \frac{f(x+h)-f(x)}{h}\in W $ in $0$ exists. The problem I face is that I don't know how to get rid of the norm in the numerator.

This question was motivated by this question, where I couldn't really understand the answer, because it assumed an answer to this question.

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I think this follows from the simple fact that, by definition of the limit, $\lim_{h\to 0}g(h)=0$ is equivalent to $\lim_{h\to 0}|g(h)|=0$. Therefore these are equivalent:

$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}=L$

$\lim_{h\to 0} \frac{f(x+h)-f(x)-Lh}{h}=0$

$\lim_{h\to 0} \frac{|f(x+h)-f(x)-Lh|}{|h|}=0$