Differentiable vector–valued functions from Euclidean space
April 24, 2024
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-dimensionalEuclidean 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
A map which may also be denoted by between two topological spaces is said to be -times continuously differentiable or if it is continuous. A topological embedding may also be called a -embedding.
Curvesedit
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 Gateaux derivative. They are fundamental to the analysis of maps between two arbitrary topological vector spaces and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.
A continuous map from a subset that is valued in a topological vector space is said to be (once or -time) differentiable if for all it is differentiable at which by definition means the following limit in exists:
where in order for this limit to even be well-defined, must be an accumulation point of If is differentiable then it is said to be continuously differentiable or if its derivative, which is the induced map is continuous. Using induction on the map is -times continuously differentiable or if its derivative is continuously differentiable, in which case the -derivative of is the map It is called smooth, or infinitely differentiable if it is -times continuously differentiable for every integer For it is called -times differentiable if it is -times continuous differentiable and is differentiable.
A continuous function from a non-empty and non-degenerate interval into a topological space is called a curve or a curve in A path in is a curve in whose domain is compact while an arc or C0-arc in is a path in that is also a topological embedding. For any a curve valued in a topological vector space is called a -embedding if it is a topological embedding and a curve such that for every where it is called a -arc if it is also a path (or equivalently, also a -arc) in addition to being a -embedding.
Differentiability on Euclidean spaceedit
The definition given above for curves are now extended from functions valued defined on subsets of to functions defined on open subsets of finite-dimensional Euclidean spaces.
Throughout, let be an open subset of where is an integer. Suppose and is a function such that with an accumulation point of Then is differentiable at [1] if there exist vectors in called the partial derivatives of at , such that
where If is differentiable at a point then it is continuous at that point.[1] If is differentiable at every point in some subset of its domain then is said to be (once or -time) differentiable in , where if the subset is not mentioned then this means that it is differentiable at every point in its domain. If is differentiable and if each of its partial derivatives is a continuous function then is said to be (once or -time) continuously differentiable or [1] For having defined what it means for a function to be (or times continuously differentiable), say that is times continuously differentiable or that is if is continuously differentiable and each of its partial derivatives is Say that is smooth, or infinitely differentiable if is for all The support of a function is the closure (taken in its domain ) of the set
a locally compact topological space, in which case can only be
Space of Ck functionsedit
For any let denote the vector space of all -valued maps defined on and let denote the vector subspace of consisting of all maps in that have compact support. Let denote and denote Give the topology of uniform convergence of the functions together with their derivatives of order on the compact subsets of [1] Suppose is a sequence of relatively compact open subsets of whose union is and that satisfy for all Suppose that is a basis of neighborhoods of the origin in Then for any integer the sets:
form a basis of neighborhoods of the origin for as and vary in all possible ways. If is a countable union of compact subsets and is a Fréchet space, then so is Note that is convex whenever is convex. If is metrizable (resp. complete, locally convex, Hausdorff) then so is [1][2] If is a basis of continuous seminorms for then a basis of continuous seminorms on is:
Space of Ck functions with support in a compact subsetedit
The definition of the topology of the space of test functions is now duplicated and generalized. For any compact subset denote the set of all in whose support lies in (in particular, if then the domain of is rather than ) and give it the subspace topology induced by [1] If is a compact space and is a Banach space, then becomes a Banach space normed by [2] Let denote For any two compact subsets the inclusion
is an embedding of TVSs and that the union of all as varies over the compact subsets of is
Space of compactly support Ck functionsedit
For any compact subset let
denote the inclusion map and endow with the strongest topology making all continuous, which is known as the final topology induced by these map. The spaces and maps form a direct system (directed by the compact subsets of ) whose limit in the category of TVSs is together with the injections [1] The spaces and maps also form a direct system (directed by the total order ) whose limit in the category of TVSs is together with the injections [1] Each embedding is an embedding of TVSs. A subset of is a neighborhood of the origin in if and only if is a neighborhood of the origin in for every compact This direct limit topology (i.e. the final topology) on is known as the canonical LF topology.
If is a Hausdorff locally convex space, is a TVS, and is a linear map, then is continuous if and only if for all compact the restriction of to is continuous.[1] The statement remains true if "all compact " is replaced with "all ".
Propertiesedit
Theorem[1] — 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 (where has its canonical LF topology).
is a continuous linear map; and furthermore, its restriction
is also continuous (where has the canonical LF topology).
Identification as a tensor productedit
Suppose henceforth that is Hausdorff. Given a function and a vector let denote the map defined by This defines a bilinear map into the space of functions whose image is contained in a finite-dimensional vector subspace of this bilinear map turns this subspace into a tensor product of and which we will denote by [1] Furthermore, if denotes the vector subspace of consisting of all functions with compact support, then is a tensor product of and [1]
If is locally compact then is dense in while if is an open subset of then is dense in [2]
Theorem — If is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product[2]
See alsoedit
Convenient vector space – locally convex vector spaces satisfying a very mild completeness conditionPages displaying wikidata descriptions as a fallback
Gateaux derivative – Generalization of the concept of directional derivative
Infinite-dimensional vector function – function whose values lie in an infinite-dimensional vector spacePages displaying wikidata descriptions as a fallback
Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). 16. Providence: American Mathematical Society. ISBN978-0-8218-1216-7. MR 0075539. OCLC 1315788.
Hogbe-Nlend, Henri; Moscatelli, V. B. (1981). Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality "topology-bornology". North-Holland Mathematics Studies. Vol. 52. Amsterdam New York New York: North Holland. ISBN978-0-08-087163-9. OCLC 316564345.
Pietsch, Albrecht (1979). Nuclear Locally Convex Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 66 (Second ed.). Berlin, New York: Springer-Verlag. ISBN978-0-387-05644-9. OCLC 539541.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. OCLC 144216834.
Ryan, Raymond A. (2002). Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. London New York: Springer. ISBN978-1-85233-437-6. OCLC 48092184.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN978-1-4612-7155-0. OCLC 840278135.
differentiable, vector, valued, functions, from, euclidean, space, mathematical, discipline, functional, analysis, differentiable, vector, valued, function, from, euclidean, space, differentiable, function, valued, topological, vector, space, whose, domains, s. 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 k displaystyle k times continuously differentiable functions on an open subset W displaystyle Omega of Euclidean space R n displaystyle mathbb R n 1 n lt displaystyle 1 leq n lt infty 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 R n displaystyle mathbb R n 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 F displaystyle mathbb F where F displaystyle mathbb F is either the real numbers R displaystyle mathbb R or the complex numbers C displaystyle mathbb C Contents 1 Continuously differentiable vector valued functions 1 1 Curves 1 2 Differentiability on Euclidean space 2 Spaces of Ck vector valued functions 2 1 Space of Ck functions 2 2 Space of Ck functions with support in a compact subset 2 3 Space of compactly support Ck functions 2 4 Properties 2 5 Identification as a tensor product 3 See also 4 Notes 5 Citations 6 ReferencesContinuously differentiable vector valued functions editA map f displaystyle f nbsp which may also be denoted by f 0 displaystyle f 0 nbsp between two topological spaces is said to be 0 displaystyle 0 nbsp times continuously differentiable or C 0 displaystyle C 0 nbsp if it is continuous A topological embedding may also be called a C 0 displaystyle C 0 nbsp embedding Curves edit 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 Gateaux derivative They are fundamental to the analysis of maps between two arbitrary topological vector spaces X Y displaystyle X to Y nbsp and so also to the analysis of TVS valued maps from Euclidean spaces which is the focus of this article A continuous map f I X displaystyle f I to X nbsp from a subset I R displaystyle I subseteq mathbb R nbsp that is valued in a topological vector space X displaystyle X nbsp is said to be once or 1 displaystyle 1 nbsp time differentiable if for all t I displaystyle t in I nbsp it is differentiable at t displaystyle t nbsp which by definition means the following limit in X displaystyle X nbsp exists f t f 1 t lim t r I r t f r f t r t lim t t h I h 0 f t h f t h displaystyle f prime t f 1 t lim stackrel r to t t neq r in I frac f r f t r t lim stackrel h to 0 t neq t h in I frac f t h f t h nbsp where in order for this limit to even be well defined t displaystyle t nbsp must be an accumulation point of I displaystyle I nbsp If f I X displaystyle f I to X nbsp is differentiable then it is said to be continuously differentiable or C 1 displaystyle C 1 nbsp if its derivative which is the induced map f f 1 I X displaystyle f prime f 1 I to X nbsp is continuous Using induction on 1 lt k N displaystyle 1 lt k in mathbb N nbsp the map f I X displaystyle f I to X nbsp is k displaystyle k nbsp times continuously differentiable or C k displaystyle C k nbsp if its k 1 th displaystyle k 1 text th nbsp derivative f k 1 I X displaystyle f k 1 I to X nbsp is continuously differentiable in which case the k th displaystyle k text th nbsp derivative of f displaystyle f nbsp is the map f k f k 1 I X displaystyle f k left f k 1 right prime I to X nbsp It is called smooth C displaystyle C infty nbsp or infinitely differentiable if it is k displaystyle k nbsp times continuously differentiable for every integer k N displaystyle k in mathbb N nbsp For k N displaystyle k in mathbb N nbsp it is called k displaystyle k nbsp times differentiable if it is k 1 displaystyle k 1 nbsp times continuous differentiable and f k 1 I X displaystyle f k 1 I to X nbsp is differentiable A continuous function f I X displaystyle f I to X nbsp from a non empty and non degenerate interval I R displaystyle I subseteq mathbb R nbsp into a topological space X displaystyle X nbsp is called a curve or a C 0 displaystyle C 0 nbsp curve in X displaystyle X nbsp A path in X displaystyle X nbsp is a curve in X displaystyle X nbsp whose domain is compact while an arc or C 0 arc in X displaystyle X nbsp is a path in X displaystyle X nbsp that is also a topological embedding For any k 1 2 displaystyle k in 1 2 ldots infty nbsp a curve f I X displaystyle f I to X nbsp valued in a topological vector space X displaystyle X nbsp is called a C k displaystyle C k nbsp embedding if it is a topological embedding and a C k displaystyle C k nbsp curve such that f t 0 displaystyle f prime t neq 0 nbsp for every t I displaystyle t in I nbsp where it is called a C k displaystyle C k nbsp arc if it is also a path or equivalently also a C 0 displaystyle C 0 nbsp arc in addition to being a C k displaystyle C k nbsp embedding Differentiability on Euclidean space edit The definition given above for curves are now extended from functions valued defined on subsets of R displaystyle mathbb R nbsp to functions defined on open subsets of finite dimensional Euclidean spaces Throughout let W displaystyle Omega nbsp be an open subset of R n displaystyle mathbb R n nbsp where n 1 displaystyle n geq 1 nbsp is an integer Suppose t t 1 t n W displaystyle t left t 1 ldots t n right in Omega nbsp and f domain f Y displaystyle f operatorname domain f to Y nbsp is a function such that t domain f displaystyle t in operatorname domain f nbsp with t displaystyle t nbsp an accumulation point of domain f displaystyle operatorname domain f nbsp Then f displaystyle f nbsp is differentiable at t displaystyle t nbsp 1 if there exist n displaystyle n nbsp vectors e 1 e n displaystyle e 1 ldots e n nbsp in Y displaystyle Y nbsp called the partial derivatives of f displaystyle f nbsp at t displaystyle t nbsp such thatlim t p domain f p t f p f t i 1 n p i t i e i p t 2 0 in Y displaystyle lim stackrel p to t t neq p in operatorname domain f frac f p f t sum i 1 n left p i t i right e i p t 2 0 text in Y nbsp where p p 1 p n displaystyle p left p 1 ldots p n right nbsp If f displaystyle f nbsp is differentiable at a point then it is continuous at that point 1 If f displaystyle f nbsp is differentiable at every point in some subset S displaystyle S nbsp of its domain then f displaystyle f nbsp is said to be once or 1 displaystyle 1 nbsp time differentiable in S displaystyle S nbsp where if the subset S displaystyle S nbsp is not mentioned then this means that it is differentiable at every point in its domain If f displaystyle f nbsp is differentiable and if each of its partial derivatives is a continuous function then f displaystyle f nbsp is said to be once or 1 displaystyle 1 nbsp time continuously differentiable or C 1 displaystyle C 1 nbsp 1 For k N displaystyle k in mathbb N nbsp having defined what it means for a function f displaystyle f nbsp to be C k displaystyle C k nbsp or k displaystyle k nbsp times continuously differentiable say that f displaystyle f nbsp is k 1 displaystyle k 1 nbsp times continuously differentiable or that f displaystyle f nbsp is C k 1 displaystyle C k 1 nbsp if f displaystyle f nbsp is continuously differentiable and each of its partial derivatives is C k displaystyle C k nbsp Say that f displaystyle f nbsp is C displaystyle C infty nbsp smooth C displaystyle C infty nbsp or infinitely differentiable if f displaystyle f nbsp is C k displaystyle C k nbsp for all k 0 1 displaystyle k 0 1 ldots nbsp The support of a function f displaystyle f nbsp is the closure taken in its domain domain f displaystyle operatorname domain f nbsp of the set x domain f f x 0 displaystyle x in operatorname domain f f x neq 0 nbsp Spaces of Ck vector valued functions editSee also Distribution mathematics In this section the space of smooth test functions and its canonical LF topology are generalized to functions valued in general complete Hausdorff locally convex topological vector spaces TVSs After this task is completed it is revealed that the topological vector space C k W Y displaystyle C k Omega Y nbsp that was constructed could up to TVS isomorphism have instead been defined simply as the completed injective tensor product C k W ϵ Y displaystyle C k Omega widehat otimes epsilon Y nbsp of the usual space of smooth test functions C k W displaystyle C k Omega nbsp with Y displaystyle Y nbsp Throughout let Y displaystyle Y nbsp be a Hausdorff topological vector space TVS let k 0 1 displaystyle k in 0 1 ldots infty nbsp and let W displaystyle Omega nbsp be either an open subset of R n displaystyle mathbb R n nbsp where n 1 displaystyle n geq 1 nbsp is an integer or else a locally compact topological space in which case k displaystyle k nbsp can only be 0 displaystyle 0 nbsp Space of Ck functions edit For any k 0 1 displaystyle k 0 1 ldots infty nbsp let C k W Y displaystyle C k Omega Y nbsp denote the vector space of all C k displaystyle C k nbsp Y displaystyle Y nbsp valued maps defined on W displaystyle Omega nbsp and let C c k W Y displaystyle C c k Omega Y nbsp denote the vector subspace of C k W Y displaystyle C k Omega Y nbsp consisting of all maps in C k W Y displaystyle C k Omega Y nbsp that have compact support Let C k W displaystyle C k Omega nbsp denote C k W F displaystyle C k Omega mathbb F nbsp and C c k W displaystyle C c k Omega nbsp denote C c k W F displaystyle C c k Omega mathbb F nbsp Give C c k W Y displaystyle C c k Omega Y nbsp the topology of uniform convergence of the functions together with their derivatives of order lt k 1 displaystyle lt k 1 nbsp on the compact subsets of W displaystyle Omega nbsp 1 Suppose W 1 W 2 displaystyle Omega 1 subseteq Omega 2 subseteq cdots nbsp is a sequence of relatively compact open subsets of W displaystyle Omega nbsp whose union is W displaystyle Omega nbsp and that satisfy W i W i 1 displaystyle overline Omega i subseteq Omega i 1 nbsp for all i displaystyle i nbsp Suppose that V a a A displaystyle left V alpha right alpha in A nbsp is a basis of neighborhoods of the origin in Y displaystyle Y nbsp Then for any integer ℓ lt k 1 displaystyle ell lt k 1 nbsp the sets U i ℓ a f C k W Y p q f p U a for all p W i and all q N n q ℓ displaystyle mathcal U i ell alpha left f in C k Omega Y left partial partial p right q f p in U alpha text for all p in Omega i text and all q in mathbb N n q leq ell right nbsp form a basis of neighborhoods of the origin for C k W Y displaystyle C k Omega Y nbsp as i displaystyle i nbsp ℓ displaystyle ell nbsp and a A displaystyle alpha in A nbsp vary in all possible ways If W displaystyle Omega nbsp is a countable union of compact subsets and Y displaystyle Y nbsp is a Frechet space then so is C W Y displaystyle C Omega Y nbsp Note that U i l a displaystyle mathcal U i l alpha nbsp is convex whenever U a displaystyle U alpha nbsp is convex If Y displaystyle Y nbsp is metrizable resp complete locally convex Hausdorff then so is C k W Y displaystyle C k Omega Y nbsp 1 2 If p a a A displaystyle p alpha alpha in A nbsp is a basis of continuous seminorms for Y displaystyle Y nbsp then a basis of continuous seminorms on C k W Y displaystyle C k Omega Y nbsp is m i l a f sup y W i q l p a p q f p displaystyle mu i l alpha f sup y in Omega i left sum q leq l p alpha left left partial partial p right q f p right right nbsp as i displaystyle i nbsp ℓ displaystyle ell nbsp and a A displaystyle alpha in A nbsp vary in all possible ways 1 Space of Ck functions with support in a compact subset edit The definition of the topology of the space of test functions is now duplicated and generalized For any compact subset K W displaystyle K subseteq Omega nbsp denote the set of all f displaystyle f nbsp in C k W Y displaystyle C k Omega Y nbsp whose support lies in K displaystyle K nbsp in particular if f C k K Y displaystyle f in C k K Y nbsp then the domain of f displaystyle f nbsp is W displaystyle Omega nbsp rather than K displaystyle K nbsp and give it the subspace topology induced by C k W Y displaystyle C k Omega Y nbsp 1 If K displaystyle K nbsp is a compact space and Y displaystyle Y nbsp is a Banach space then C 0 K Y displaystyle C 0 K Y nbsp becomes a Banach space normed by f sup w W f w displaystyle f sup omega in Omega f omega nbsp 2 Let C k K displaystyle C k K nbsp denote C k K F displaystyle C k K mathbb F nbsp For any two compact subsets K L W displaystyle K subseteq L subseteq Omega nbsp the inclusionIn K L C k K Y C k L Y displaystyle operatorname In K L C k K Y to C k L Y nbsp is an embedding of TVSs and that the union of all C k K Y displaystyle C k K Y nbsp as K displaystyle K nbsp varies over the compact subsets of W displaystyle Omega nbsp is C c k W Y displaystyle C c k Omega Y nbsp Space of compactly support Ck functions edit For any compact subset K W displaystyle K subseteq Omega nbsp letIn K C k K Y C c k W Y displaystyle operatorname In K C k K Y to C c k Omega Y nbsp denote the inclusion map and endow C c k W Y displaystyle C c k Omega Y nbsp with the strongest topology making all In K displaystyle operatorname In K nbsp continuous which is known as the final topology induced by these map The spaces C k K Y displaystyle C k K Y nbsp and maps In K 1 K 2 displaystyle operatorname In K 1 K 2 nbsp form a direct system directed by the compact subsets of W displaystyle Omega nbsp whose limit in the category of TVSs is C c k W Y displaystyle C c k Omega Y nbsp together with the injections In K displaystyle operatorname In K nbsp 1 The spaces C k W i Y displaystyle C k left overline Omega i Y right nbsp and maps In W i W j displaystyle operatorname In overline Omega i overline Omega j nbsp also form a direct system directed by the total order N displaystyle mathbb N nbsp whose limit in the category of TVSs is C c k W Y displaystyle C c k Omega Y nbsp together with the injections In W i displaystyle operatorname In overline Omega i nbsp 1 Each embedding In K displaystyle operatorname In K nbsp is an embedding of TVSs A subset S displaystyle S nbsp of C c k W Y displaystyle C c k Omega Y nbsp is a neighborhood of the origin in C c k W Y displaystyle C c k Omega Y nbsp if and only if S C k K Y displaystyle S cap C k K Y nbsp is a neighborhood of the origin in C k K Y displaystyle C k K Y nbsp for every compact K W displaystyle K subseteq Omega nbsp This direct limit topology i e the final topology on C c W displaystyle C c infty Omega nbsp is known as the canonical LF topology If Y displaystyle Y nbsp is a Hausdorff locally convex space T displaystyle T nbsp is a TVS and u C c k W Y T displaystyle u C c k Omega Y to T nbsp is a linear map then u displaystyle u nbsp is continuous if and only if for all compact K W displaystyle K subseteq Omega nbsp the restriction of u displaystyle u nbsp to C k K Y displaystyle C k K Y nbsp is continuous 1 The statement remains true if all compact K W displaystyle K subseteq Omega nbsp is replaced with all K W i displaystyle K overline Omega i nbsp Properties edit Theorem 1 Let m displaystyle m nbsp be a positive integer and let D displaystyle Delta nbsp be an open subset of R m displaystyle mathbb R m nbsp Given ϕ C k W D displaystyle phi in C k Omega times Delta nbsp for any y D displaystyle y in Delta nbsp let ϕ y W F displaystyle phi y Omega to mathbb F nbsp be defined by ϕ y x ϕ x y displaystyle phi y x phi x y nbsp and let I k ϕ D C k W displaystyle I k phi Delta to C k Omega nbsp be defined by I k ϕ y ϕ y displaystyle I k phi y phi y nbsp ThenI C W D C D C W displaystyle I infty C infty Omega times Delta to C infty Delta C infty Omega nbsp is a surjective isomorphism of TVSs Furthermore its restriction I C c W D C c W D C c D C c W displaystyle I infty big vert C c infty left Omega times Delta right C c infty Omega times Delta to C c infty left Delta C c infty Omega right nbsp is an isomorphism of TVSs where C c W D displaystyle C c infty left Omega times Delta right nbsp has its canonical LF topology Theorem 1 Let Y displaystyle Y nbsp be a Hausdorff locally convex topological vector space and for every continuous linear form y Y displaystyle y prime in Y nbsp and every f C W Y displaystyle f in C infty Omega Y nbsp let J y f W F displaystyle J y prime f Omega to mathbb F nbsp be defined by J y f p y f p displaystyle J y prime f p y prime f p nbsp ThenJ y C W Y C W displaystyle J y prime C infty Omega Y to C infty Omega nbsp is a continuous linear map and furthermore its restriction J y C c W Y C c W Y C W displaystyle J y prime big vert C c infty Omega Y C c infty Omega Y to C infty Omega nbsp is also continuous where C c W Y displaystyle C c infty Omega Y nbsp has the canonical LF topology Identification as a tensor product edit Suppose henceforth that Y displaystyle Y nbsp is Hausdorff Given a function f C k W displaystyle f in C k Omega nbsp and a vector y Y displaystyle y in Y nbsp let f y displaystyle f otimes y nbsp denote the map f y W Y displaystyle f otimes y Omega to Y nbsp defined by f y p f p y displaystyle f otimes y p f p y nbsp This defines a bilinear map C k W Y C k W Y displaystyle otimes C k Omega times Y to C k Omega Y nbsp into the space of functions whose image is contained in a finite dimensional vector subspace of Y displaystyle Y nbsp this bilinear map turns this subspace into a tensor product of C k W displaystyle C k Omega nbsp and Y displaystyle Y nbsp which we will denote by C k W Y displaystyle C k Omega otimes Y nbsp 1 Furthermore if C c k W Y displaystyle C c k Omega otimes Y nbsp denotes the vector subspace of C k W Y displaystyle C k Omega otimes Y nbsp consisting of all functions with compact support then C c k W Y displaystyle C c k Omega otimes Y nbsp is a tensor product of C c k W displaystyle C c k Omega nbsp and Y displaystyle Y nbsp 1 If X displaystyle X nbsp is locally compact then C c 0 W Y displaystyle C c 0 Omega otimes Y nbsp is dense in C 0 W X displaystyle C 0 Omega X nbsp while if X displaystyle X nbsp is an open subset of R n displaystyle mathbb R n nbsp then C c W Y displaystyle C c infty Omega otimes Y nbsp is dense in C k W X displaystyle C k Omega X nbsp 2 Theorem If Y displaystyle Y nbsp is a complete Hausdorff locally convex space then C k W Y displaystyle C k Omega Y nbsp is canonically isomorphic to the injective tensor product C k W ϵ Y displaystyle C k Omega widehat otimes epsilon Y nbsp 2 See also editConvenient vector space locally convex vector spaces satisfying a very mild completeness conditionPages displaying wikidata descriptions as a fallback Crinkled arc Differentiation in Frechet spaces Frechet derivative Derivative defined on normed spaces Gateaux derivative Generalization of the concept of directional derivative Infinite dimensional vector function function whose values lie in an infinite dimensional vector spacePages displaying wikidata descriptions as a fallback Injective tensor productNotes editCitations edit a b c d e f g h i j k l m n Treves 2006 pp 412 419 a b c d Treves 2006 pp 446 451 References editDiestel Joe 2008 The Metric Theory of Tensor Products Grothendieck s Resume Revisited Vol 16 Providence R I American Mathematical Society ISBN 9781470424831 OCLC 185095773 Dubinsky Ed 1979 The Structure of Nuclear Frechet Spaces Lecture Notes in Mathematics Vol 720 Berlin New York Springer Verlag ISBN 978 3 540 09504 0 OCLC 5126156 Grothendieck Alexander 1955 Produits Tensoriels Topologiques et Espaces Nucleaires Topological Tensor Products and Nuclear Spaces Memoirs of the American Mathematical Society Series in French 16 Providence American Mathematical Society ISBN 978 0 8218 1216 7 MR 0075539 OCLC 1315788 Grothendieck Alexander 1973 Topological Vector Spaces Translated by Chaljub Orlando New York Gordon and Breach Science Publishers ISBN 978 0 677 30020 7 OCLC 886098 Hogbe Nlend Henri Moscatelli V B 1981 Nuclear and Conuclear Spaces Introductory Course on Nuclear and Conuclear Spaces in the Light of the Duality topology bornology North Holland Mathematics Studies Vol 52 Amsterdam New York New York North Holland ISBN 978 0 08 087163 9 OCLC 316564345 Khaleelulla S M 1982 Counterexamples in Topological Vector Spaces Lecture Notes in Mathematics Vol 936 Berlin Heidelberg New York Springer Verlag ISBN 978 3 540 11565 6 OCLC 8588370 Pietsch Albrecht 1979 Nuclear Locally Convex Spaces Ergebnisse der Mathematik und ihrer Grenzgebiete Vol 66 Second ed Berlin New York Springer Verlag ISBN 978 0 387 05644 9 OCLC 539541 Narici Lawrence Beckenstein Edward 2011 Topological Vector Spaces Pure and applied mathematics Second ed Boca Raton FL CRC Press ISBN 978 1584888666 OCLC 144216834 Ryan Raymond A 2002 Introduction to Tensor Products of Banach Spaces Springer Monographs in Mathematics London New York Springer ISBN 978 1 85233 437 6 OCLC 48092184 Schaefer Helmut H Wolff Manfred P 1999 Topological Vector Spaces GTM Vol 8 Second ed New York NY Springer New York Imprint Springer ISBN 978 1 4612 7155 0 OCLC 840278135 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 Wong Yau Chuen 1979 Schwartz Spaces Nuclear Spaces and Tensor Products Lecture Notes in Mathematics Vol 726 Berlin New York Springer Verlag ISBN 978 3 540 09513 2 OCLC 5126158 Retrieved from https en wikipedia org w index php title Differentiable vector valued functions from Euclidean space amp oldid 1189942289, wikipedia, wiki, book, books, library,