Differentiable vector-valued functions from Euclidean space

In the field of Functional Analysis, it is possible to generalize the notion of derivative to infinite dimensional topological vector spaces 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 more well behaved. This article presents the theory k-times continuously differentiable functions on an open subset $\Omega$ of Euclidean space $\mathbb {R} ^{n}$ ($1\leq n<\infty$). All vector spaces will be assumed to be over the field $\mathbb {F}$, where $\mathbb {F}$ is either the real numbers $\mathbb {R}$ or the complex numbers $\mathbb {C}$.

Continuously differentiable vector-valued functions

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

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

and let $Y$ be a topological vector space (TVS).

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

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

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

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

Spaces of C^k vector-valued functions

Space of C^k functions

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

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

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

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

as i, l, and $\alpha \in A$ vary in all possible ways.

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

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

Space of compactly support C^k functions

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

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

Properties

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

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

Identification as a tensor product

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

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

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

See also

• Injective tensor product

References

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