Differentiable vector-valued functions from Euclidean space

In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset of some finite-dimensional Euclidean space.

It is possible to generalize the notion of derivative to functions whose domain and codomain are subsets of arbitrary topological vector spaces (TVSs) in multiple ways.

But when the domain of a TVS-valued function is a subset of a finite-dimensional Euclidean space then many of these notions become logically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more well-behaved compared to the general case.

This article presents the theory of

k

{\displaystyle k}

-times continuously differentiable functions on an open subset

Ω

{\displaystyle \Omega }

of Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

(

1

≤

n

<

∞

{\displaystyle 1\leq n<\infty }

), which is an important special case of differentiation between arbitrary TVSs.

This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is TVS isomorphic to Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by restricting it to finite-dimensional vector subspaces.

All vector spaces will be assumed to be over the field

F

,

{\displaystyle \mathbb {F} ,}

where

F

{\displaystyle \mathbb {F} }

is either the real numbers

R

{\displaystyle \mathbb {R} }

or the complex numbers

C

.

{\displaystyle \mathbb {C} .}

Continuously differentiable vector-valued functions

A map

f

,

{\displaystyle f,}

which may also be denoted by

f

(

0

)

,

{\displaystyle f^{(0)},}

between two topological spaces is said to be

0

{\displaystyle 0}

-times continuously differentiable or

C

0

{\displaystyle C^{0}}

if it is continuous. A topological embedding may also be called a

C

0

{\displaystyle C^{0}}

-embedding.

Curves

Differentiable curves are an important special case of differentiable vector-valued (i.e. TVS-valued) functions which, in particular, are used in the definition of the Gateaux derivative.