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In the mathematical discipline offunctional analysis, adifferentiable vector-valued function from Euclidean space is adifferentiable function valued in atopological vector space (TVS) whosedomains is a subset of somefinite-dimensionalEuclidean space. It is possible to generalize the notion ofderivative to functions whose domain and codomain are subsets of arbitrarytopological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-valued function is a subset of a finite-dimensionalEuclidean space then many of these notions becomelogically equivalent resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also morewell-behaved compared to the general case. This article presents the theory of${\displaystyle k}$-times continuously differentiable functions on an open subset${\displaystyle \mathrm{\Omega }}$ of Euclidean space${\displaystyle {\mathbb{R}}^n}$ (), which is an important special case ofdifferentiation between arbitrary TVSs. This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space isTVS isomorphic to Euclidean space${\displaystyle {\mathbb{R}}^n}$ so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS byrestricting it to finite-dimensional vector subspaces.

All vector spaces will be assumed to be over the field${\displaystyle \mathbb{F},}$ where${\displaystyle \mathbb{F}}$ is either thereal numbers${\displaystyle \mathbb{R}}$ or thecomplex numbers${\displaystyle \mathbb{C}.}$

A map${\displaystyle f,}$ which may also be denoted by${\displaystyle f^{(0)},}$ between twotopological spaces is said to be${\displaystyle 0}$-times continuously differentiable or${\displaystyle C^0}$ if it is continuous. Atopological embedding may also be called a${\displaystyle C^0}$-embedding.

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 theGateaux derivative. They are fundamental to the analysis of maps between two arbitrarytopological vector spaces${\displaystyle X\to Y}$ and so also to the analysis of TVS-valued maps fromEuclidean spaces, which is the focus of this article.

A continuous map${\displaystyle f:I\to X}$ from a subset${\displaystyle I\subseteq \mathbb{R}}$ that is valued in atopological vector space${\displaystyle X}$ is said to be (once or${\displaystyle 1}$-time)differentiable if for all${\displaystyle t\in I,}$ it isdifferentiable at${\displaystyle t,}$ which by definition means the followinglimit in${\displaystyle X}$ exists:$${\displaystyle f^{\mathrm{\prime }}(t):=f^{(1)}(t):=\underset{\stackrel{r\to t}{t\ne r\in I}}{lim}\frac{f(r)-f(t)}{r-t}=\underset{\stackrel{h\to 0}{t\ne t+h\in I}}{lim}\frac{f(t+h)-f(t)}{h}}$$ where in order for this limit to even be well-defined,${\displaystyle t}$ must be anaccumulation point of${\displaystyle I.}$ If${\displaystyle f:I\to X}$ is differentiable then it is said to becontinuously differentiable or${\displaystyle C^1}$ if itsderivative, which is the induced map${\displaystyle f^{\mathrm{\prime }}=f^{(1)}:I\to X,}$ is continuous. Using induction on the map${\displaystyle f:I\to X}$ is${\displaystyle k}$-times continuously differentiable or${\displaystyle C^k}$ if its${\displaystyle k-1^{\text{th}}}$ derivative${\displaystyle f^{(k-1)}:I\to X}$ is continuously differentiable, in which case the${\displaystyle k^{\text{th}}}$-derivative of${\displaystyle f}$ is the map${\displaystyle f^{(k)}:={\left(f^{(k-1)}\right)}^{\mathrm{\prime }}:I\to X.}$ It is calledsmooth,${\displaystyle C^{\mathrm{\infty }},}$ orinfinitely differentiable if it is${\displaystyle k}$-times continuously differentiable for every integer${\displaystyle k\in \mathbb{N}.}$ For${\displaystyle k\in \mathbb{N},}$ it is called${\displaystyle k}$-times differentiable if it is${\displaystyle k-1}$-times continuous differentiable and${\displaystyle f^{(k-1)}:I\to X}$ is differentiable.

A continuous function${\displaystyle f:I\to X}$ from a non-empty and non-degenerate interval${\displaystyle I\subseteq \mathbb{R}}$ into atopological space${\displaystyle X}$ is called acurve or a${\displaystyle C^0}$ curve in${\displaystyle X.}$ Apath in${\displaystyle X}$ is a curve in${\displaystyle X}$ whose domain is compact while anarc orC⁰-arc in${\displaystyle X}$ is a path in${\displaystyle X}$ that is also atopological embedding. For any${\displaystyle k\in \{1,2,\dots ,\mathrm{\infty }\},}$ a curve${\displaystyle f:I\to X}$ valued in a topological vector space${\displaystyle X}$ is called a${\displaystyle C^k}$-embedding if it is atopological embedding and a${\displaystyle C^k}$ curve such that${\displaystyle f^{\mathrm{\prime }}(t)\ne 0}$ for every${\displaystyle t\in I,}$ where it is called a${\displaystyle C^k}$-arc if it is also a path (or equivalently, also a${\displaystyle C^0}$-arc) in addition to being a${\displaystyle C^k}$-embedding.

The definition given above for curves are now extended from functions valued defined on subsets of${\displaystyle \mathbb{R}}$ to functions defined on open subsets of finite-dimensionalEuclidean spaces.

Throughout, let${\displaystyle \mathrm{\Omega }}$ be an open subset of${\displaystyle {\mathbb{R}}^n,}$ where${\displaystyle n\ge 1}$ is an integer. Suppose${\displaystyle t=\left(t_1,\dots ,t_n\right)\in \mathrm{\Omega }}$ and${\displaystyle f:\mathrm{domain}f\to Y}$ is a function such that${\displaystyle t\in \mathrm{domain}f}$ with${\displaystyle t}$ an accumulation point of${\displaystyle \mathrm{domain}f.}$ Then${\displaystyle f}$ isdifferentiable at${\displaystyle t}$^[1] if there exist${\displaystyle n}$ vectors${\displaystyle e_1,\dots ,e_n}$ in${\displaystyle Y,}$ called thepartial derivatives of${\displaystyle f}$ at${\displaystyle t}$, such that$${\displaystyle \underset{\stackrel{p\to t}{t\ne p\in \mathrm{domain}f}}{lim}\frac{f(p)-f(t)-\sum _{i=1}^n\left(p_i-t_i\right)e_i}{\Vert p-t{\Vert }_2}=0\text{ in }Y}$$where${\displaystyle p=\left(p_1,\dots ,p_n\right).}$ If${\displaystyle f}$ is differentiable at a point then it is continuous at that point.^[1] If${\displaystyle f}$ is differentiable at every point in some subset${\displaystyle S}$ of its domain then${\displaystyle f}$ is said to be (once or${\displaystyle 1}$-time)differentiable in${\displaystyle S}$, where if the subset${\displaystyle S}$ is not mentioned then this means that it is differentiable at every point in its domain. If${\displaystyle f}$ is differentiable and if each of its partial derivatives is a continuous function then${\displaystyle f}$ is said to be (once or${\displaystyle 1}$-time)continuously differentiable or${\displaystyle C^1.}$^[1] For${\displaystyle k\in \mathbb{N},}$ having defined what it means for a function${\displaystyle f}$ to be${\displaystyle C^k}$ (or${\displaystyle k}$ times continuously differentiable), say that${\displaystyle f}$ is${\displaystyle k+1}$ times continuously differentiable or that${\displaystyle f}$ is${\displaystyle C^{k+1}}$ if${\displaystyle f}$ is continuously differentiable and each of its partial derivatives is${\displaystyle C^k.}$ Say that${\displaystyle f}$ is${\displaystyle C^{\mathrm{\infty }},}$smooth,${\displaystyle C^{\mathrm{\infty }},}$ orinfinitely differentiable if${\displaystyle f}$ is${\displaystyle C^k}$ for all${\displaystyle k=0,1,\dots .}$ Thesupport of a function${\displaystyle f}$ is theclosure (taken in its domain${\displaystyle \mathrm{domain}f}$) of the set${\displaystyle \{x\in \mathrm{domain}f:f(x)\ne 0\}.}$

In this section, thespace of smooth test functions and its canonical LF-topology are generalized to functions valued in generalcomplete Hausdorff locally convextopological vector spaces (TVSs). After this task is completed, it is revealed that the topological vector space${\displaystyle C^k(\mathrm{\Omega };Y)}$ that was constructed could (up to TVS-isomorphism) have instead been defined simply as the completedinjective tensor product${\displaystyle C^k(\mathrm{\Omega }){\hat{\otimes }}_{ϵ}Y}$ of the usualspace of smooth test functions${\displaystyle C^k(\mathrm{\Omega })}$ with${\displaystyle Y.}$

Throughout, let${\displaystyle Y}$ be a Hausdorfftopological vector space (TVS), let${\displaystyle k\in \{0,1,\dots ,\mathrm{\infty }\},}$ and let${\displaystyle \mathrm{\Omega }}$ be either:

1. an open subset of${\displaystyle {\mathbb{R}}^n,}$ where${\displaystyle n\ge 1}$ is an integer, or else
2. alocally compact topological space, in which case${\displaystyle k}$ can only be${\displaystyle 0.}$

For any${\displaystyle k=0,1,\dots ,\mathrm{\infty },}$ let${\displaystyle C^k(\mathrm{\Omega };Y)}$ denote the vector space of all${\displaystyle C^k}$${\displaystyle Y}$-valued maps defined on${\displaystyle \mathrm{\Omega }}$ and let${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ denote the vector subspace of${\displaystyle C^k(\mathrm{\Omega };Y)}$ consisting of all maps in${\displaystyle C^k(\mathrm{\Omega };Y)}$ that have compact support. Let${\displaystyle C^k(\mathrm{\Omega })}$ denote${\displaystyle C^k(\mathrm{\Omega };\mathbb{F})}$ and${\displaystyle C_c^k(\mathrm{\Omega })}$ denote${\displaystyle C_c^k(\mathrm{\Omega };\mathbb{F}).}$ Give${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ the topology of uniform convergence of the functions together with their derivatives of order on the compact subsets of${\displaystyle \mathrm{\Omega }.}$^[1] Suppose${\displaystyle {\mathrm{\Omega }}_1\subseteq {\mathrm{\Omega }}_2\subseteq \cdots }$ is a sequence ofrelatively compact open subsets of${\displaystyle \mathrm{\Omega }}$ whose union is${\displaystyle \mathrm{\Omega }}$ and that satisfy${\displaystyle \overline{{\mathrm{\Omega }}_i}\subseteq {\mathrm{\Omega }}_{i+1}}$ for all${\displaystyle i.}$Suppose that${\displaystyle {\left(V_{\alpha }\right)}_{\alpha \in A}}$ is a basis of neighborhoods of the origin in${\displaystyle Y.}$ Then for any integer the sets:$${\displaystyle {\mathcal{U}}_{i,\ell ,\alpha }:=\left\{f\in C^k(\mathrm{\Omega };Y):{\left(\mathrm{\partial }/\mathrm{\partial }p\right)}^{q}f(p)\in U_{\alpha }\text{ for all }p\in {\mathrm{\Omega }}_i\text{ and all }q\in {\mathbb{N}}^n,|q|\le \ell \right\}}$$form a basis of neighborhoods of the origin for${\displaystyle C^k(\mathrm{\Omega };Y)}$ as${\displaystyle i,}$${\displaystyle \ell ,}$ and${\displaystyle \alpha \in A}$ vary in all possible ways. If${\displaystyle \mathrm{\Omega }}$ is a countable union of compact subsets and${\displaystyle Y}$ is aFréchet space, then so is${\displaystyle C^{(}\mathrm{\Omega };Y).}$ Note that${\displaystyle {\mathcal{U}}_{i,l,\alpha }}$ is convex whenever${\displaystyle U_{\alpha }}$ is convex. If${\displaystyle Y}$ ismetrizable (resp.complete,locally convex,Hausdorff) then so is${\displaystyle C^k(\mathrm{\Omega };Y).}$^[1][2] If${\displaystyle (p_{\alpha }{)}_{\alpha \in A}}$ is a basis of continuous seminorms for${\displaystyle Y}$ then a basis of continuous seminorms on${\displaystyle C^k(\mathrm{\Omega };Y)}$ is:$${\displaystyle {\mu }_{i,l,\alpha }(f):=\underset{y\in {\mathrm{\Omega }}_i}{sup}\left(\sum _{|q|\le l}{p}_{\alpha }\left({\left(\mathrm{\partial }/\mathrm{\partial }p\right)}^{q}f(p)\right)\right)}$$as${\displaystyle i,}$${\displaystyle \ell ,}$ and${\displaystyle \alpha \in A}$ vary in all possible ways.^[1]

The definition of the topology of thespace of test functions is now duplicated and generalized. For any compact subset${\displaystyle K\subseteq \mathrm{\Omega },}$ denote the set of all${\displaystyle f}$ in${\displaystyle C^k(\mathrm{\Omega };Y)}$ whose support lies in${\displaystyle K}$ (in particular, if${\displaystyle f\in C^k(K;Y)}$ then the domain of${\displaystyle f}$ is${\displaystyle \mathrm{\Omega }}$ rather than${\displaystyle K}$) and give it the subspace topology induced by${\displaystyle C^k(\mathrm{\Omega };Y).}$^[1] If${\displaystyle K}$ is a compact space and${\displaystyle Y}$ is a Banach space, then${\displaystyle C^0(K;Y)}$ becomes a Banach space normed by${\displaystyle \Vert f\Vert :=\underset{\omega \in \mathrm{\Omega }}{sup}\Vert f(\omega )\Vert .}$^[2] Let${\displaystyle C^k(K)}$ denote${\displaystyle C^k(K;\mathbb{F}).}$ For any two compact subsets${\displaystyle K\subseteq L\subseteq \mathrm{\Omega },}$ the inclusion$${\displaystyle {\mathrm{In}}_K^L:C^k(K;Y)\to C^k(L;Y)}$$ is an embedding of TVSs and that the union of all${\displaystyle C^k(K;Y),}$ as${\displaystyle K}$ varies over the compact subsets of${\displaystyle \mathrm{\Omega },}$ is${\displaystyle C_c^k(\mathrm{\Omega };Y).}$

For any compact subset${\displaystyle K\subseteq \mathrm{\Omega },}$ let$${\displaystyle {\mathrm{In}}_K:C^k(K;Y)\to C_c^k(\mathrm{\Omega };Y)}$$ denote the inclusion map and endow${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ with the strongest topology making all${\displaystyle {\mathrm{In}}_K}$ continuous, which is known as thefinal topology induced by these map. The spaces${\displaystyle C^k(K;Y)}$ and maps${\displaystyle {\mathrm{In}}_{K_1}^{K_2}}$ form adirect system (directed by the compact subsets of${\displaystyle \mathrm{\Omega }}$) whose limit in the category of TVSs is${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ together with the injections${\displaystyle {\mathrm{In}}_K.}$^[1] The spaces${\displaystyle C^k\left(\overline{{\mathrm{\Omega }}_i};Y\right)}$ and maps${\displaystyle {\mathrm{In}}_{\overline{{\mathrm{\Omega }}_i}}^{\overline{{\mathrm{\Omega }}_j}}}$ also form adirect system (directed by the total order${\displaystyle \mathbb{N}}$) whose limit in the category of TVSs is${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ together with the injections${\displaystyle {\mathrm{In}}_{\overline{{\mathrm{\Omega }}_i}}.}$^[1] Each embedding${\displaystyle {\mathrm{In}}_K}$ is an embedding of TVSs. A subset${\displaystyle S}$ of${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ is a neighborhood of the origin in${\displaystyle C_c^k(\mathrm{\Omega };Y)}$ if and only if${\displaystyle S\cap C^k(K;Y)}$ is a neighborhood of the origin in${\displaystyle C^k(K;Y)}$ for every compact${\displaystyle K\subseteq \mathrm{\Omega }.}$ This direct limit topology (i.e. the final topology) on${\displaystyle C_c^{\mathrm{\infty }}(\mathrm{\Omega })}$ is known as thecanonical LF topology.

If${\displaystyle Y}$ is a Hausdorff locally convex space,${\displaystyle T}$ is a TVS, and${\displaystyle u:C_c^k(\mathrm{\Omega };Y)\to T}$ is a linear map, then${\displaystyle u}$ is continuous if and only if for all compact${\displaystyle K\subseteq \mathrm{\Omega },}$ the restriction of${\displaystyle u}$ to${\displaystyle C^k(K;Y)}$ is continuous.^[1] The statement remains true if "all compact${\displaystyle K\subseteq \mathrm{\Omega }}$" is replaced with "all${\displaystyle K:={\overline{\mathrm{\Omega }}}_i}$".

Suppose henceforth that${\displaystyle Y}$ is Hausdorff. Given a function${\displaystyle f\in C^k(\mathrm{\Omega })}$ and a vector${\displaystyle y\in Y,}$ let${\displaystyle f\otimes y}$ denote the map${\displaystyle f\otimes y:\mathrm{\Omega }\to Y}$ defined by${\displaystyle (f\otimes y)(p)=f(p)y.}$ This defines a bilinear map${\displaystyle \otimes :C^k(\mathrm{\Omega })\times Y\to C^k(\mathrm{\Omega };Y)}$ into the space of functions whose image is contained in a finite-dimensional vector subspace of${\displaystyle Y;}$ this bilinear map turns this subspace into a tensor product of${\displaystyle C^k(\mathrm{\Omega })}$ and${\displaystyle Y,}$ which we will denote by${\displaystyle C^k(\mathrm{\Omega })\otimes Y.}$^[1] Furthermore, if${\displaystyle C_c^k(\mathrm{\Omega })\otimes Y}$ denotes the vector subspace of${\displaystyle C^k(\mathrm{\Omega })\otimes Y}$ consisting of all functions with compact support, then${\displaystyle C_c^k(\mathrm{\Omega })\otimes Y}$ is a tensor product of${\displaystyle C_c^k(\mathrm{\Omega })}$ and${\displaystyle Y.}$^[1]

If${\displaystyle X}$ is locally compact then${\displaystyle C_c^0(\mathrm{\Omega })\otimes Y}$ is dense in${\displaystyle C^0(\mathrm{\Omega };X)}$ while if${\displaystyle X}$ is an open subset of${\displaystyle {\mathbb{R}}^n}$ then${\displaystyle C_c^{\mathrm{\infty }}(\mathrm{\Omega })\otimes Y}$ is dense in${\displaystyle C^k(\mathrm{\Omega };X).}$^[2]

• Convenient vector space
• Crinkled arc
• Differentiation in Fréchet spaces
• Fréchet derivative – Derivative defined on normed spaces
• Gateaux derivative – Generalization of the concept of directional derivative
• Infinite-dimensional vector function – Whose values lie in an infinite-dimensional vector space
• Injective tensor product

• Diestel, Joe (2008).The Metric Theory of Tensor Products: Grothendieck's Résumé Revisited. Vol. 16. Providence, R.I.:American Mathematical Society.ISBN 9781470424831.OCLC 185095773.
• Dubinsky, Ed (1979).The Structure of Nuclear Fréchet 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 Nucléaires" [Topological Tensor Products and Nuclear Spaces].Memoirs of the American Mathematical Society Series (in French).16. Providence: American Mathematical Society.MR 0075539.OCLC 9308061.
• 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.
• Trèves, François (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.

• Banach spaces
• Differential calculus
• Euclidean geometry
• Functions and mappings
• Generalizations of the derivative
• Topological vector spaces

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