Definition:Differentiable Mapping/Function With Values in Normed Space

Definition
Let $U \subset \R$ be an open set.

Let $\struct {X, \norm \cdot_X}$ be a normed vector space.

A function $f : U \to X$ is differentiable at $x \in U$ there exists $f'(x) \in X$ such that:


 * $\ds \lim_{h \mathop \to 0} \Big\|\frac {\map f {x+h} - \map f {x}} h-f'(x)\Big\|_{X}=0$