Differentiable vector-valued functions from Euclidean space

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]

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 ${\displaystyle C_{c}^{k}\left(\Omega ;Y\right).}$

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 ${\displaystyle K:={\overline {\Omega }}_{i}}$".