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 (strongly) differentiable at $x \in U$ if and only if there exists $\map {f'} x \in X$ such that:

$\ds \lim_{h \mathop \to 0} \norm {\frac {\map f {x + h} - \map f x} h - \map {f'} x}_X = 0$

Moreover, $f$ is called (strongly) differentiable if it is differentiable at every point of $U$.

Sources

• 1996: E. Hille and R.S. Phillips: Functional Analysis and Semi-Groups (Revised ed.): Chapter $\text {III}$: Vector-Valued Functions: $3.2$. Some properties of vector-valued functions. Definition $3.2.3$