Differentiable vector-valued functions from Euclidean space

All vector spaces will be assumed to be over the field $F,$ where $F$ is either the real numbers $R$ or the complex numbers $C .$

Continuously differentiable vector-valued functions

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

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 $X \to Y$ and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.

A continuous map $f : I \to X$ from a subset $I \subseteq R$ that is valued in a topological vector space $X$ is said to be (once or $1$ -time) differentiable if for all $t \in I,$ it is differentiable at $t,$ which by definition means the following limit in $X$ exists: $f^{'} (t) := f^{(1)} (t) := lim_{\overset{r \to t}{t \neq r \in I}} \frac{f (r) - f (t)}{r - t} = lim_{\overset{h \to 0}{t \neq t + h \in I}} \frac{f (t + h) - f (t)}{h}$ where in order for this limit to even be well-defined, $t$ must be an accumulation point of $I .$ If $f : I \to X$ is differentiable then it is said to be continuously differentiable or $C^{1}$ if its derivative, which is the induced map $f^{'} = f^{(1)} : I \to X,$ is continuous. Using induction on $1 < k \in N,$ the map $f : I \to X$ is $k$ -times continuously differentiable or $C^{k}$ if its $k - 1^{th}$ derivative $f^{(k - 1)} : I \to X$ is continuously differentiable, in which case the $k^{th}$ -derivative of $f$ is the map $f^{(k)} := {(f^{(k - 1)})}^{'} : I \to X .$ It is called smooth, $C^{\infty},$ or infinitely differentiable if it is $k$ -times continuously differentiable for every integer $k \in N .$ For $k \in N,$ it is called $k$ -times differentiable if it is $k - 1$ -times continuous differentiable and $f^{(k - 1)} : I \to X$ is differentiable.

A continuous function $f : I \to X$ from a non-empty and non-degenerate interval $I \subseteq R$ into a topological space $X$ is called a curve or a $C^{0}$ curve in $X .$ A path in $X$ is a curve in $X$ whose domain is compact while an arc or C⁰-arc in $X$ is a path in $X$ that is also a topological embedding. For any $k \in {1, 2, \dots, \infty},$ a curve $f : I \to X$ valued in a topological vector space $X$ is called a $C^{k}$ -embedding if it is a topological embedding and a $C^{k}$ curve such that $f^{'} (t) \neq 0$ for every $t \in I,$ where it is called a $C^{k}$ -arc if it is also a path (or equivalently, also a $C^{0}$ -arc) in addition to being a $C^{k}$ -embedding.

Differentiability on Euclidean space

The definition given above for curves are now extended from functions valued defined on subsets of $R$ to functions defined on open subsets of finite-dimensional Euclidean spaces.

Throughout, let $Ω$ be an open subset of $R^{n},$ where $n \geq 1$ is an integer. Suppose $t = (t_{1}, \dots, t_{n}) \in Ω$ and $f : domain f \to Y$ is a function such that $t \in domain f$ with $t$ an accumulation point of $domain f .$ Then $f$ is differentiable at $t$ if there exist $n$ vectors $e_{1}, \dots, e_{n}$ in $Y,$ called the partial derivatives of $f$ at $t$ , such that $lim_{\overset{p \to t}{t \neq p \in domain f}} \frac{f (p) - f (t) - \sum_{i = 1}^{n} (p_{i} - t_{i}) e_{i}}{‖ p - t ‖_{2}} = 0 in Y$ where $p = (p_{1}, \dots, p_{n}) .$ If $f$ is differentiable at a point then it is continuous at that point. If $f$ is differentiable at every point in some subset $S$ of its domain then $f$ is said to be (once or $1$ -time) differentiable in $S$ , where if the subset $S$ is not mentioned then this means that it is differentiable at every point in its domain. If $f$ is differentiable and if each of its partial derivatives is a continuous function then $f$ is said to be (once or $1$ -time) continuously differentiable or $C^{1} .$ For $k \in N,$ 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, $C^{\infty},$ or infinitely differentiable if $f$ is $C^{k}$ for all $k = 0, 1, \dots .$ The support of a function $f$ is the closure (taken in its domain $domain f$ ) of the set ${x \in domain f : f (x) \neq 0} .$

One more chapter is free with a Lably account.

Spaces of Ck vector-valued functions

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

Throughout, let $Y$ be a Hausdorff topological vector space (TVS), let $k \in {0, 1, \dots, \infty},$ and let $Ω$ be either:

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

Space of Ck functions

For any $k = 0, 1, \dots, \infty,$ let $C^{k} (Ω; Y)$ denote the vector space of all $C^{k}$ $Y$ -valued maps defined on $Ω$ and let $C_{c}^{k} (Ω; Y)$ denote the vector subspace of $C^{k} (Ω; Y)$ consisting of all maps in $C^{k} (Ω; Y)$ that have compact support. Let $C^{k} (Ω)$ denote $C^{k} (Ω; F)$ and $C_{c}^{k} (Ω)$ denote $C_{c}^{k} (Ω; F) .$ Give $C_{c}^{k} (Ω; Y)$ the topology of uniform convergence of the functions together with their derivatives of order $< k + 1$ on the compact subsets of $Ω .$ Suppose $Ω_{1} \subseteq Ω_{2} \subseteq \dots$ is a sequence of relatively compact open subsets of $Ω$ whose union is $Ω$ and that satisfy $\bar{Ω_{i}} \subseteq Ω_{i + 1}$ for all $i .$ Suppose that ${(V_{α})}_{α \in A}$ is a basis of neighborhoods of the origin in $Y .$ Then for any integer $ℓ < k + 1,$ the sets: $U_{i, ℓ, α} := {f \in C^{k} (Ω; Y) : {(\partial / \partial p)}^{q} f (p) \in U_{α} for all p \in Ω_{i} and all q \in N^{n}, | q | \leq ℓ}$ form a basis of neighborhoods of the origin for $C^{k} (Ω; Y)$ as $i,$ $ℓ,$ and $α \in A$ vary in all possible ways. If $Ω$ is a countable union of compact subsets and $Y$ is a Fréchet space, then so is $C^{(} Ω; Y) .$ Note that $U_{i, l, α}$ is convex whenever $U_{α}$ is convex. If $Y$ is metrizable (resp. complete, locally convex, Hausdorff) then so is $C^{k} (Ω; Y) .$ If $(p_{α})_{α \in A}$ is a basis of continuous seminorms for $Y$ then a basis of continuous seminorms on $C^{k} (Ω; Y)$ is: $μ_{i, l, α} (f) := sup_{y \in Ω_{i}} (\sum_{| q | \leq l} p_{α} ({(\partial / \partial p)}^{q} f (p)))$ as $i,$ $ℓ,$ and $α \in A$ vary in all possible ways.

Space of Ck 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 $K \subseteq Ω,$ denote the set of all $f$ in $C^{k} (Ω; Y)$ whose support lies in $K$ (in particular, if $f \in C^{k} (K; Y)$ then the domain of $f$ is $Ω$ rather than $K$ ) and give it the subspace topology induced by $C^{k} (Ω; Y) .$ If $K$ is a compact space and $Y$ is a Banach space, then $C^{0} (K; Y)$ becomes a Banach space normed by $‖ f ‖ := sup_{ω \in Ω} ‖ f (ω) ‖ .$ Let $C^{k} (K)$ denote $C^{k} (K; F) .$ For any two compact subsets $K \subseteq L \subseteq Ω,$ the inclusion ${In}_{K}^{L} : C^{k} (K; Y) \to C^{k} (L; Y)$ is an embedding of TVSs and that the union of all $C^{k} (K; Y),$ as $K$ varies over the compact subsets of $Ω,$ is $C_{c}^{k} (Ω; Y) .$

Space of compactly support Ck functions

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

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

Properties

Theorem—Let $m$ be a positive integer and let $Δ$ be an open subset of $R^{m} .$ Given $ϕ \in C^{k} (Ω \times Δ),$ for any $y \in Δ$ let $ϕ_{y} : Ω \to F$ be defined by $ϕ_{y} (x) = ϕ (x, y)$ and let $I_{k} (ϕ) : Δ \to C^{k} (Ω)$ be defined by $I_{k} (ϕ) (y) := ϕ_{y} .$ Then $I_{\infty} : C^{\infty} (Ω \times Δ) \to C^{\infty} (Δ; C^{\infty} (Ω))$ is a surjective isomorphism of TVSs. Furthermore, its restriction $I_{\infty} |_{C_{c}^{\infty} (Ω \times Δ)} : C_{c}^{\infty} (Ω \times Δ) \to C_{c}^{\infty} (Δ; C_{c}^{\infty} (Ω))$ is an isomorphism of TVSs (where $C_{c}^{\infty} (Ω \times Δ)$ has its canonical LF topology).

Theorem—Let $Y$ be a Hausdorff locally convex topological vector space and for every continuous linear form $y^{'} \in Y$ and every $f \in C^{\infty} (Ω; Y),$ let $J_{y^{'}} (f) : Ω \to F$ be defined by $J_{y^{'}} (f) (p) = y^{'} (f (p)) .$ Then $J_{y^{'}} : C^{\infty} (Ω; Y) \to C^{\infty} (Ω)$ is a continuous linear map; and furthermore, its restriction $J_{y^{'}} |_{C_{c}^{\infty} (Ω; Y)} : C_{c}^{\infty} (Ω; Y) \to C^{\infty} (Ω)$ is also continuous (where $C_{c}^{\infty} (Ω; Y)$ has the canonical LF topology).

Identification as a tensor product

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

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

Theorem—If $Y$ is a complete Hausdorff locally convex space, then $C^{k} (Ω; Y)$ is canonically isomorphic to the injective tensor product $C^{k} (Ω) {\hat{\otimes}}_{ϵ} Y .$