Differentiable vector-valued functions from Euclidean space

In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to infinite dimensional topological vector spaces (TVSs) in multiple ways. But when the domain of TVS-value functions is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of k-times continuously differentiable functions on an open subset ${\displaystyle \Omega }$ of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (${\displaystyle 1\leq n<\infty }$), which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field ${\displaystyle \mathbb {F} ,}$ where ${\displaystyle \mathbb {F} }$ is either the real numbers ${\displaystyle \mathbb {R} }$ or the complex numbers ${\displaystyle \mathbb {C} .}$

Continuously differentiable vector-valued functions

Throughout, let ${\displaystyle k\in \{0,1,\ldots ,\infty \}}$ and let ${\displaystyle \Omega }$ be either:

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

and let ${\displaystyle Y}$ be a topological vector space (TVS).

Suppose ${\displaystyle p^{0}=\left(p_{1}^{0},\ldots ,p_{n}^{0}\right)\in \Omega }$ and ${\displaystyle f:\operatorname {Dom} f\to Y}$ is a function such that ${\displaystyle p^{0}\in \operatorname {Dom} f}$ with ${\displaystyle p^{0}}$ a limit point of ${\displaystyle \operatorname {Dom} f.}$ Then f is differentiable at ${\displaystyle p^{0}}$[1] if there exist n vectors ${\displaystyle e_{1},\ldots ,e_{n}}$ in Y, called the partial derivatives of f, such that

${\displaystyle \lim _{p\to p^{0},p\in \operatorname {Dom} f}{\frac {f(p)-f(p^{0})-\sum _{i=1}^{n}\left(p_{i}-p_{i}^{0}\right)e_{i}}{\|p-p^{0}\|_{2}}}=0}$ in Y

where ${\displaystyle p=(p_{1},\ldots ,p_{n}).}$

If f is differentiable at a point then it is continuous at that point.[1] Say that f is ${\displaystyle C^{0}}$ if it is continuous. If f is differentiable at every point in some set ${\displaystyle S\subseteq \Omega }$ then we say that f is differentiable in S. If f is differentiable at every point of its domain and if each of its partial derivatives is a continuous function then we say that f is continuously differentiable or ${\displaystyle C^{1}.}$[1] Having defined what it means for a function f to be ${\displaystyle C^{k}}$ (or k times continuously differentiable), say that f is k + 1 times continuously differentiable or that f is ${\displaystyle C^{k+1}}$ if f is continuously differentiable and each of its partial derivatives is ${\displaystyle C^{k}.}$ Say that f is ${\displaystyle C^{\infty },}$ smooth, or infinitely differentiable if f is ${\displaystyle C^{k}}$ for all ${\displaystyle k=0,1,\ldots .}$ If ${\displaystyle f:\Omega \to Y}$ is any function then its support is the closure (in ${\displaystyle \Omega }$) of the set ${\displaystyle \{x\in \operatorname {Dom} f:f(x)\neq 0\}.}$

Spaces of C^k vector-valued functions

Space of C^k functions

For any ${\displaystyle k=0,1,\ldots ,\infty ,}$ let ${\displaystyle C^{k}(\Omega ;Y)}$ denote the vector space of all ${\displaystyle C^{k}}$ Y-valued maps defined on ${\displaystyle \Omega }$ and let ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ denote the vector subspace of ${\displaystyle C^{k}\left(\Omega ;Y\right)}$ consisting of all maps in ${\displaystyle C^{k}(\Omega ;Y)}$ that have compact support. Let ${\displaystyle C^{k}(\Omega )}$ denote ${\displaystyle C^{k}\left(\Omega$ ;\mathbb {F} \right)} and ${\displaystyle C_{c}^{k}\left(\Omega \right)}$ denote ${\displaystyle C_{c}^{k}(\Omega$ ;\mathbb {F} ).} Give ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ the topology of uniform convergence of the functions together with their derivatives of order < k + 1 on the compact subsets of ${\displaystyle \Omega .}$[1] Suppose ${\displaystyle \Omega _{1}\subseteq \Omega _{2}\subseteq \cdots }$ is a sequence of relatively compact open subsets of ${\displaystyle \Omega }$ whose union is ${\displaystyle \Omega }$ and that satisfy ${\displaystyle {\overline {\Omega _{i}}}\subseteq \Omega _{i+1}}$ for all i. Suppose that ${\displaystyle (V_{\alpha })_{\alpha \in A}}$ is a basis of neighborhoods of the origin in Y. Then for any integer ${\displaystyle \ell <k+1,}$ the sets:

${\displaystyle {\mathcal {U}}_{i,\ell ,\alpha }:=\left\{f\in C^{k}(\Omega ;Y):\left(\partial /\partial p\right)^{q}f(p)\in U_{\alpha }{\text{ for all }}p\in \Omega _{i}{\text{ and all }}q\in \mathbb {N} ^{n},|q|\leq \ell \right\}}$

form a basis of neighborhoods of the origin for ${\displaystyle C^{k}(\Omega ;Y)}$ as i, l, and ${\displaystyle \alpha \in A}$ vary in all possible ways. If ${\displaystyle \Omega }$ is a countable union of compact subsets and Y is a Fréchet space, then so is ${\displaystyle C^{(}\Omega ;Y).}$ Note that ${\displaystyle {\mathcal {U}}_{i,l,\alpha }}$ is convex whenever ${\displaystyle U_{\alpha }}$ is convex. If Y is metrizable (resp. complete, locally convex, Hausdorff) then so is ${\displaystyle C^{k}\left(\Omega ;Y\right).}$[1][2] If ${\displaystyle (p_{\alpha })_{\alpha \in A}}$ is a basis of continuous seminorms for Y then a basis of continuous seminorms on ${\displaystyle C^{k}(\Omega ;Y)}$ is:

${\displaystyle \mu _{i,l,\alpha }(f):=\sup _{y\in \Omega _{i}}\left(\sum _{|q|\leq l}p_{\alpha }\left(\left(\partial /\partial p\right)^{q}f(p)\right)\right)}$

as i, l, and ${\displaystyle \alpha \in A}$ vary in all possible ways.[1]

If ${\displaystyle \Omega }$ is a compact space and Y is a Banach space, then ${\displaystyle C^{0}\left(\Omega ;Y\right)}$ becomes a Banach space normed by ${\displaystyle \|f\|:=\sup _{\omega \in \Omega }\|f(\omega )\|.}$[2]

Space of C^k functions with support in a compact subset

We now duplicate the definition of the topology of the space of test functions. For any compact subset ${\displaystyle K\subseteq \Omega ,}$ let ${\displaystyle C^{k}\left(K;Y\right)}$ denote the set of all f in ${\displaystyle C^{k}\left(\Omega ;Y\right)}$ whose support lies in K (in particular, if ${\displaystyle f\in C^{k}\left(K;Y\right)}$ then the domain of f is ${\displaystyle \Omega }$ rather than K) and give ${\displaystyle C^{k}\left(K;Y\right)}$ the subspace topology induced by ${\displaystyle C^{k}\left(\Omega ;Y\right).}$[1] Let ${\displaystyle C^{k}\left(K\right)}$ denote ${\displaystyle C^{k}\left(K;\mathbb {F} \right).}$ Note that for any two compact subsets ${\displaystyle K_{1}\subseteq K_{2}\subseteq \Omega ,}$ the natural inclusion ${\displaystyle \operatorname {In} _{K_{1}}^{K_{2}}:C^{k}\left(K_{1};Y\right)\to C^{k}\left(K_{2};Y\right)}$ is an embedding of TVSs and that the union of all ${\displaystyle C^{k}\left(K;Y\right),}$ as K varies over the compact subsets of ${\displaystyle \Omega ,}$ is ${\displaystyle C_{c}^{k}\left(\Omega ;Y\right).}$

Space of compactly support C^k functions

For any compact subset ${\displaystyle K\subseteq \Omega ,}$ let ${\displaystyle \operatorname {In} _{K}:C^{k}(K;Y)\to C_{c}^{k}(\Omega ;Y)}$ be the natural inclusion and give ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ the strongest topology making all ${\displaystyle \operatorname {In} _{K}}$ continuous. The spaces ${\displaystyle C^{k}(K;Y)}$ and maps ${\displaystyle \operatorname {In} _{K_{1}}^{K_{2}}}$ form a direct system (directed by the compact subsets of ${\displaystyle \Omega }$) whose limit in the category of TVSs is ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ together with the natural injections ${\displaystyle \operatorname {In} _{K}.}$[1] The spaces ${\displaystyle C^{k}\left({\overline {\Omega _{i}}};Y\right)}$ and maps ${\displaystyle \operatorname {In} _{\overline {\Omega _{i}}}^{\overline {\Omega _{j}}}}$ also form a direct system (directed by the total order ${\displaystyle \mathbb {N} }$) whose limit in the category of TVSs is ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ together with the natural injections ${\displaystyle \operatorname {In} _{\overline {\Omega _{i}}}.}$[1] Each natural embedding ${\displaystyle \operatorname {In} _{K}}$ is an embedding of TVSs. A subset S of ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ is a neighborhood of the origin in ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ if and only if ${\displaystyle S\cap C^{k}(K;Y)}$ is a neighborhood of the origin in ${\displaystyle C^{k}\left(K;Y\right)}$ for every compact ${\displaystyle K\subseteq \Omega .}$ This direct limit topology on ${\displaystyle C_{c}^{\infty }(\Omega )}$ is known as the canonical LF topology.

If Y is a Hausdorff locally convex space, T is a TVS, and ${\displaystyle u:C_{c}^{k}(\Omega ;Y)\to T}$ is a linear map, then u is continuous if and only if for all compact ${\displaystyle K\subseteq \Omega ,}$ the restriction of u to ${\displaystyle C^{k}(K;Y)}$ is continuous.[1] One replace "all compact ${\displaystyle K\subseteq \Omega }$" with "all ${\displaystyle K:={\overline {\Omega }}_{i}}$".

Properties

Theorem[1] — Let m be a positive integer and let ${\displaystyle \Delta }$ be an open subset of ${\displaystyle \mathbb {R} ^{m}.}$ Given ${\displaystyle \phi \in C^{k}(\Omega \times \Delta ),}$ for any ${\displaystyle y\in \Delta }$ let ${\displaystyle \phi _{y}:\Omega \to \mathbb {F} }$ be defined by ${\displaystyle \phi _{y}(x)=\phi (x,y)}$; and let ${\displaystyle I_{k}(\phi ):\Delta \to C^{k}(\Omega )}$ be defined by ${\displaystyle I_{k}(\phi )(y):=\phi _{y}.}$ Then ${\displaystyle I_{\infty }:C^{\infty }(\Omega \times \Delta )\to C^{\infty }(\Delta ;C^{\infty }(\Omega ))}$ is a (surjective) isomorphism of TVSs. Furthermore, the restriction ${\displaystyle I_{\infty }{\big \vert }_{C_{c}^{\infty }\left(\Omega \times \Delta \right)}:C_{c}^{\infty }(\Omega \times \Delta )\to C_{c}^{\infty }\left(\Delta ;C_{c}^{\infty }(\Omega )\right)}$ is an isomorphism of TVSs when ${\displaystyle C_{c}^{\infty }\left(\Omega \times \Delta \right)}$ has its canonical LF topology.

Theorem[3] — Let Y be a Hausdorff locally convex space. For every continuous linear form ${\displaystyle y^{\prime }\in Y}$ and every ${\displaystyle f\in C^{\infty }(\Omega ;Y),}$ let ${\displaystyle J_{y^{\prime }}(f):\Omega \to \mathbb {F} }$ be defined by ${\displaystyle J_{y^{\prime }}(f)(p)=y^{\prime }(f(p)).}$ Then ${\displaystyle J_{y^{\prime }}:C^{\infty }(\Omega ;Y)\to C^{\infty }(\Omega )}$ is a continuous linear map; and furthermore, the restriction ${\displaystyle J_{y^{\prime }}{\big \vert }_{C_{c}^{\infty }(\Omega ;Y)}:C_{c}^{\infty }(\Omega ;Y)\to C^{\infty }(\Omega )}$ is also continuous (where ${\displaystyle C_{c}^{\infty }(\Omega ;Y)}$ has the canonical LF topology).

Identification as a tensor product

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

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

Theorem — If Y is a complete Hausdorff locally convex space, then ${\displaystyle C^{k}\left(\Omega ;Y\right)}$ is canonically isomorphic to the injective tensor product ${\displaystyle C^{k}\left(\Omega \right){\widehat {\otimes }}_{\epsilon }Y.}$[4]

See also

• Fréchet derivative
• Injective tensor product

References

1. Trèves 2006, pp. 412–419.
2. Trèves 2006, pp. 446–451.
3. Trèves 2006, pp. 412-419.
4. Trèves 2006, pp. 446-451.

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