Differentiable vector-valued functions from Euclidean space: Difference between revisions

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In the mathematical discipline of [[functional analysis]], it'''differentiable isvector-valued possiblefunctions tofrom generalizeEuclidean thespace''' notionare of[[differentiable]] [[Derivativetopological (mathematics)vector space|derivativeTVS]]-valued tofunctions arbitrarywhose (i.e.[[Domain of a function|domains]] are subset of [[Dimension (vector space)|infinite finite-dimensional]]) [[topological vectorEuclidean space]]s (TVSs) in multiple ways.

It is possible to generalize the notion of [[Derivative (mathematics)|derivative]] to functions whose ___domain and codomain are subsets of arbitrary [[topological vector space]]s (TVSs) in multiple ways.

But when the ___domain of a TVS-valuevalued function is a subset of a finite-dimensional [[Euclidean space]] then the numbermany of generalizationsthese ofnotions thebecome derivative[[logically isequivalent]] resulting in a much more limited number of generalizations of the derivative and moreover, derivatives that are more [[well -behaved]] compared to the more general case.

This article presents the theory of <math>k</math>-times continuously differentiable functions on an open subset <math>\Omega</math> of Euclidean space <math>\R^n</math> (<math>1 \leq n < \infty</math>), which is an important special case of [[Differentiation (mathematics)|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 isomorphism|TVS isomorphic]] to Euclidean space <math>\R^n</math> so that, for example, this special case can be applied to any function whose ___domain is an arbitrary Hausdorff TVS by [[Restriction of a function|restricting it]] to finite-dimensional vector subspaces.

All vector spaces will be assumed to be over the field <math>\mathbb{F},</math> where <math>\mathbb{F}</math> is either the [[real numbers]] <math>\R</math> or the [[complex numbers]] <math>\C.</math>