| dbo:abstract
|
- 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 -times continuously differentiable functions on an open subset of Euclidean space, 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 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 where is either the real numbers or the complex numbers (en)
|
| dbo:wikiPageID
| |
| dbo:wikiPageLength
|
- 21085 (xsd:nonNegativeInteger)
|
| dbo:wikiPageRevisionID
| |
| dbo:wikiPageWikiLink
| |
| dbp:mathStatement
|
- Let be a positive integer and let be an open subset of
Given for any let be defined by and let be defined by
Then
is a surjective isomorphism of TVSs.
Furthermore, its restriction
is an isomorphism of TVSs . (en)
- If is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product (en)
- Let be a Hausdorff locally convex topological vector space and for every continuous linear form and every let be defined by
Then
is a continuous linear map;
and furthermore, its restriction
is also continuous . (en)
|
| dbp:name
| |
| dbp:wikiPageUsesTemplate
| |
| dcterms:subject
| |
| rdf:type
| |
| rdfs:comment
|
- 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 -times continuou (en)
|
| rdfs:label
|
- Differentiable vector–valued functions from Euclidean space (en)
|
| rdfs:seeAlso
| |
| owl:sameAs
| |
| prov:wasDerivedFrom
| |
| foaf:isPrimaryTopicOf
| |
| is dbo:wikiPageRedirects
of | |
| is dbo:wikiPageWikiLink
of | |
| is foaf:primaryTopic
of | |
