About: Differentiable vector–valued functions from Euclidean space

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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 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

Attributes	Values
rdf:type	Thing
rdfs:label	Differentiable vector–valued functions from Euclidean space (en)
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:seeAlso	Distribution (mathematics)
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Subject	Banach spaces Differential calculus Euclidean geometry Topological vector spaces Functions and mappings Generalizations of the derivative
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Differentiable_vector–valued_functions_from_Euclidean_space?oldid=1063668563&ns=0
Wikipage page ID	63944266 (xsd:integer)
page length (characters) of wiki page	21079 (xsd:nonNegativeInteger)
Wikipage revision ID	1063668563 (xsd:integer)
Link from a Wikipage to another Wikipage	Locally compact space Locally convex topological vector space Logical equivalence Support (mathematics) Banach spaces Differential calculus Euclidean geometry Topological vector spaces Functions and mappings Generalizations of the derivative Differentiable function Dimension (vector space) Direct limit Distribution (mathematics) Domain of a function Metrizable topological vector space Closure (topology) Complete topological vector space Complex number Restriction (mathematics) Embedding Final topology Fréchet space Functional analysis Gateaux derivative Derivative Hausdorff space Injective tensor product Euclidean space Topological space Topological vector space Path (topology) Pathological (mathematics) Limit of a function Limit point Real number Relatively compact subspace
has 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)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Differentiable_vector–valued_functions_from_Euclidean_space
is rdfs:seeAlso of	Convenient vector space
is Wikipage redirect of	Differentiable vector-valued functions from Euclidean space Continuously differentiable vector-valued functions from Euclidean space
is Link from a Wikipage to another Wikipage of	Differentiable vector-valued functions from Euclidean space Complete topological vector space Continuously differentiable vector-valued functions from Euclidean space Injective tensor product
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Differentiable_vector–valued_functions_from_Euclidean_space

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