Curves

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 ${\displaystyle X\to Y}$ and so also to the analysis of TVS-valued maps from Euclidean 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 a topological vector space ${\displaystyle X}$ is said to be (once or ${\displaystyle 1}$-time) differentiable if for all ${\displaystyle t\in I,}$ it is differentiable at ${\displaystyle t,}$ which by definition means the following limit in ${\displaystyle X}$ exists:

$${\displaystyle f^{\prime }(t):=f^{(1)}(t):=\lim _{\stackrel {r\to t}{t\neq r\in I}}{\frac {f(r)-f(t)}{r-t}}=\lim _{\stackrel {h\to 0}{t\neq t+h\in I}}{\frac {f(t+h)-f(t)}{h}}}$$

where in order for this limit to even be well-defined, ${\displaystyle t}$ must be an accumulation point of ${\displaystyle I.}$ If ${\displaystyle f:I\to X}$ is differentiable then it is said to be continuously differentiable or ${\displaystyle C^{1}}$ if its derivative, which is the induced map ${\displaystyle f^{\prime }=f^{(1)}:I\to X,}$ is continuous. Using induction on ${\displaystyle 1<k\in \mathbb {N} ,}$ 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 ${\displaystyle f^{(k)}:=\left(f^{(k-1)}\right)^{\prime }:I\to X.}$ It is called smooth, ${\displaystyle C^{\infty },}$ or infinitely 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 a topological space ${\displaystyle X}$ is called a curve or a ${\displaystyle C^{0}}$ curve in ${\displaystyle X.}$ A path in ${\displaystyle X}$ is a curve in ${\displaystyle X}$ whose domain is compact while an arc or C⁰-arc in ${\displaystyle X}$ is a path in ${\displaystyle X}$ that is also a topological embedding. For any ${\displaystyle k\in \{1,2,\ldots ,\infty \},}$ 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 a topological embedding and a ${\displaystyle C^{k}}$ curve such that ${\displaystyle f^{\prime }(t)\neq 0}$ 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.

Differentiability on Euclidean space

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-dimensional Euclidean spaces.

Throughout, let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n},}$ where ${\displaystyle n\geq 1}$ is an integer. Suppose ${\displaystyle t=\left(t_{1},\ldots ,t_{n}\right)\in \Omega }$ and ${\displaystyle f:\operatorname {domain} f\to Y}$ is a function such that ${\displaystyle t\in \operatorname {domain} f}$ with ${\displaystyle t}$ an accumulation point of ${\displaystyle \operatorname {domain} f.}$ Then ${\displaystyle f}$ is differentiable at ${\displaystyle t}$^[1] if there exist ${\displaystyle n}$ vectors ${\displaystyle e_{1},\ldots ,e_{n}}$ in ${\displaystyle Y,}$ called the partial derivatives of ${\displaystyle f}$ at ${\displaystyle t}$, such that

$${\displaystyle \lim _{\stackrel {p\to t}{t\neq p\in \operatorname {domain} f}}{\frac {f(p)-f(t)-\sum _{i=1}^{n}\left(p_{i}-t_{i}\right)e_{i}}{\|p-t\|_{2}}}=0{\text{ in }}Y}$$

where ${\displaystyle p=\left(p_{1},\ldots ,p_{n}\right).}$ 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 ${\displaystyle C^{\infty },}$ smooth, ${\displaystyle C^{\infty },}$ or infinitely differentiable if ${\displaystyle f}$ is ${\displaystyle C^{k}}$ for all ${\displaystyle k=0,1,\ldots .}$ The support of a function ${\displaystyle f}$ is the closure (taken in its domain ${\displaystyle \operatorname {domain} f}$) of the set ${\displaystyle \{x\in \operatorname {domain} f:f(x)\neq 0\}.}$

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 ${\displaystyle C^{k}}$ ${\displaystyle Y}$-valued maps defined on ${\displaystyle \Omega }$ and let ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ denote the vector subspace of ${\displaystyle C^{k}(\Omega ;Y)}$ consisting of all maps in ${\displaystyle C^{k}(\Omega ;Y)}$ that have compact support. Let ${\displaystyle C^{k}(\Omega )}$ denote ${\displaystyle C^{k}(\Omega$ ;\mathbb {F} )} and ${\displaystyle C_{c}^{k}(\Omega )}$ 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 ${\displaystyle <k+1}$ 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 ${\displaystyle i.}$ 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 ${\displaystyle \ell <k+1,}$ 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 ${\displaystyle i,}$ ${\displaystyle \ell ,}$ and ${\displaystyle \alpha \in A}$ vary in all possible ways. If ${\displaystyle \Omega }$ is a countable union of compact subsets and ${\displaystyle Y}$ 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 ${\displaystyle Y}$ is metrizable (resp. complete, locally convex, Hausdorff) then so is ${\displaystyle C^{k}(\Omega ;Y).}$^[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 ${\displaystyle C^{k}(\Omega ;Y)}$ 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 ${\displaystyle i,}$ ${\displaystyle \ell ,}$ and ${\displaystyle \alpha \in A}$ vary in all possible ways.^[1]

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

The definition of the topology of the space of test functions is now duplicated and generalized. For any compact subset ${\displaystyle K\subseteq \Omega ,}$ denote the set of all ${\displaystyle f}$ in ${\displaystyle C^{k}(\Omega ;Y)}$ whose support lies in ${\displaystyle K}$ (in particular, if ${\displaystyle f\in C^{k}(K;Y)}$ then the domain of ${\displaystyle f}$ is ${\displaystyle \Omega }$ rather than ${\displaystyle K}$) and give it the subspace topology induced by ${\displaystyle C^{k}(\Omega ;Y).}$^[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 ${\displaystyle \|f\|:=\sup _{\omega \in \Omega }\|f(\omega )\|.}$^[2] Let ${\displaystyle C^{k}(K)}$ denote ${\displaystyle C^{k}(K;\mathbb {F} ).}$ For any two compact subsets ${\displaystyle K\subseteq L\subseteq \Omega ,}$ the inclusion

$${\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K;Y)\to C^{k}(L;Y)}$$

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 ${\displaystyle \Omega ,}$ is ${\displaystyle C_{c}^{k}(\Omega ;Y).}$

Space of compactly support C^k functions

For any compact subset ${\displaystyle K\subseteq \Omega ,}$ let

$${\displaystyle \operatorname {In} _{K}:C^{k}(K;Y)\to C_{c}^{k}(\Omega ;Y)}$$

denote the inclusion map and endow ${\displaystyle C_{c}^{k}(\Omega ;Y)}$ with the strongest topology making all ${\displaystyle \operatorname {In} _{K}}$ continuous, which is known as the final topology induced by these map. 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 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 injections ${\displaystyle \operatorname {In} _{\overline {\Omega _{i}}}.}$^[1] Each embedding ${\displaystyle \operatorname {In} _{K}}$ is an embedding of TVSs. A subset ${\displaystyle S}$ 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 ${\displaystyle C^{k}(K;Y)}$ for every compact ${\displaystyle K\subseteq \Omega .}$ This direct limit topology (i.e. the final topology) on ${\displaystyle C_{c}^{\infty }(\Omega )}$ is known as the canonical LF topology.

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

Properties

Theorem^[1] — Let ${\displaystyle m}$ 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, its 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 (where ${\displaystyle C_{c}^{\infty }\left(\Omega \times \Delta \right)}$ has its canonical LF topology).

Theorem^[1] — Let ${\displaystyle Y}$ be a Hausdorff locally convex topological vector space and 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, its 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 ${\displaystyle Y}$ is Hausdorff. Given a function ${\displaystyle f\in C^{k}(\Omega )}$ and a vector ${\displaystyle y\in Y,}$ let ${\displaystyle f\otimes y}$ denote the map ${\displaystyle f\otimes y:\Omega \to Y}$ defined by ${\displaystyle (f\otimes y)(p)=f(p)y.}$ This defines a bilinear map ${\displaystyle \otimes :C^{k}(\Omega )\times Y\to C^{k}(\Omega ;Y)}$ 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 ${\displaystyle C^{k}(\Omega )}$ and ${\displaystyle Y,}$ which we will denote by ${\displaystyle C^{k}(\Omega )\otimes Y.}$^[1] Furthermore, if ${\displaystyle C_{c}^{k}(\Omega )\otimes Y}$ denotes the vector subspace of ${\displaystyle C^{k}(\Omega )\otimes Y}$ consisting of all functions with compact support, then ${\displaystyle C_{c}^{k}(\Omega )\otimes Y}$ is a tensor product of ${\displaystyle C_{c}^{k}(\Omega )}$ and ${\displaystyle Y.}$^[1]

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