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

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{{Short description|Differentiable function in functional analysis}}

{{one source|date=January 2025}}

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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 [[Domain of a function|domains]] is a subset of some [[Dimension (vector space)|finite-dimensional]] [[Euclidean space]].

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.

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=== 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 [[GâteauxGateaux derivative]]. They are fundamental to the analysis of maps between two arbitrary [[topological vector space]]s <math>X \to Y</math> and so also to the analysis of TVS-valued maps from [[Euclidean space]]s, which is the focus of this article.

A continuous map <math>f : I \to X</math> from a subset <math>I \subseteq \mathbb{R}</math> that is valued in a [[topological vector space]] <math>X</math> is said to be ('''{{em|once}}''' or '''{{em|<math>1</math>-time}}''') '''{{em|differentiable}}''' if for all <math>t \in I,</math> it is '''{{em|differentiable at <math>t,</math>}}''' which by definition means the following [[Limit of a function#Functions on topological spaces|limit in <math>X</math>]] exists:

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* {{annotated link|Convenient vector space}}

* {{annotated link|Crinkled arc}}

* {{annotated link|Differentiation in Fréchet spaces}}

* {{annotated link|Fréchet derivative}}

* {{annotated link|Gateaux derivative}}

* {{annotated link|Infinite-dimensional vector function}}

* {{annotated link|Injective tensor product}}

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* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn|Wong|1979|p=}} -->

{{Functional Analysis in topological vector spaces}}

{{Topological vector spaces}}

{{TopologicalVectorSpaces}}

{{Functional analysis}}

{{AnalysisInTopologicalVectorSpaces}}

<!--- Categories --->

{{DEFAULTSORT:Differentiable vector-valued functions from Euclidean space}}

[[Category:Functions and mappings]]

[[Category:Banach spaces]]

[[Category:Differential calculus]]

[[Category:Euclidean geometry]]

[[Category:Functions and mappings]]

[[Category:Generalizations of the derivative]]

[[Category:Topological vector spaces]]