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

Revision as of 21:41, 1 November 2021

Revision as of 21:43, 1 November 2021

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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|Differentiation in Fréchet spaces}}

* {{annotated link|Fréchet derivative}}

* {{annotated link|Gateaux derivative}}

* {{annotated link|Injective tensor product}}