1. Continuously Differentiable Vector-Valued Functions

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

1.1. 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 [math]\displaystyle{ X \to Y }[/math] and so also to the analysis of TVS-valued maps from Euclidean spaces, which is the focus of this article.

A continuous map [math]\displaystyle{ f : I \to X }[/math] from a subset [math]\displaystyle{ I \subseteq \mathbb{R} }[/math] that is valued in a topological vector space [math]\displaystyle{ X }[/math] is said to be (once or [math]\displaystyle{ 1 }[/math]-time) differentiable if for all [math]\displaystyle{ t \in I, }[/math] it is differentiable at [math]\displaystyle{ t, }[/math] which by definition means the following limit in [math]\displaystyle{ X }[/math] exists: [math]\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} }[/math] where in order for this limit to even be well-defined, [math]\displaystyle{ t }[/math] must be an accumulation point of [math]\displaystyle{ I. }[/math] If [math]\displaystyle{ f : I \to X }[/math] is differentiable then it is said to be continuously differentiable or [math]\displaystyle{ C^1 }[/math] if its derivative, which is the induced map [math]\displaystyle{ f^{\prime} = f^{(1)} : I \to X, }[/math] is continuous. Using induction on [math]\displaystyle{ 1 \lt k \in \N, }[/math] the map [math]\displaystyle{ f : I \to X }[/math] is [math]\displaystyle{ k }[/math]-times continuously differentiable or [math]\displaystyle{ C^k }[/math] if its [math]\displaystyle{ k-1^{\text{th}} }[/math] derivative [math]\displaystyle{ f^{(k-1)} : I \to X }[/math] is continuously differentiable, in which case the [math]\displaystyle{ k^{\text{th}} }[/math]-derivative of [math]\displaystyle{ f }[/math] is the map [math]\displaystyle{ f^{(k)} := \left(f^{(k-1)}\right)^{\prime} : I \to X. }[/math] It is called smooth, [math]\displaystyle{ C^\infty, }[/math] or infinitely differentiable if it is [math]\displaystyle{ k }[/math]-times continuously differentiable for every integer [math]\displaystyle{ k \in \N. }[/math] For [math]\displaystyle{ k \in \N, }[/math] it is called [math]\displaystyle{ k }[/math]-times differentiable if it is [math]\displaystyle{ k-1 }[/math]-times continuous differentiable and [math]\displaystyle{ f^{(k-1)} : I \to X }[/math] is differentiable.

A continuous function [math]\displaystyle{ f : I \to X }[/math] from a non-empty and non-degenerate interval [math]\displaystyle{ I \subseteq \R }[/math] into a topological space [math]\displaystyle{ X }[/math] is called a curve or a [math]\displaystyle{ C^0 }[/math] curve in [math]\displaystyle{ X. }[/math] A path in [math]\displaystyle{ X }[/math] is a curve in [math]\displaystyle{ X }[/math] whose domain is compact while an arc or C⁰-arc in [math]\displaystyle{ X }[/math] is a path in [math]\displaystyle{ X }[/math] that is also a topological embedding. For any [math]\displaystyle{ k \in \{ 1, 2, \ldots, \infty \}, }[/math] a curve [math]\displaystyle{ f : I \to X }[/math] valued in a topological vector space [math]\displaystyle{ X }[/math] is called a [math]\displaystyle{ C^k }[/math]-embedding if it is a topological embedding and a [math]\displaystyle{ C^k }[/math] curve such that [math]\displaystyle{ f^{\prime}(t) \neq 0 }[/math] for every [math]\displaystyle{ t \in I, }[/math] where it is called a [math]\displaystyle{ C^k }[/math]-arc if it is also a path (or equivalently, also a [math]\displaystyle{ C^0 }[/math]-arc) in addition to being a [math]\displaystyle{ C^k }[/math]-embedding.

1.2. Differentiability on Euclidean Space

The definition given above for curves are now extended from functions valued defined on subsets of [math]\displaystyle{ \R }[/math] to functions defined on open subsets of finite-dimensional Euclidean spaces.

Throughout, let [math]\displaystyle{ \Omega }[/math] be an open subset of [math]\displaystyle{ \R^n, }[/math] where [math]\displaystyle{ n \geq 1 }[/math] is an integer. Suppose [math]\displaystyle{ t = \left( t_1, \ldots, t_n \right) \in \Omega }[/math] and [math]\displaystyle{ f : \operatorname{domain} f \to Y }[/math] is a function such that [math]\displaystyle{ t \in \operatorname{domain} f }[/math] with [math]\displaystyle{ t }[/math] an accumulation point of [math]\displaystyle{ \operatorname{domain} f. }[/math] Then [math]\displaystyle{ f }[/math] is differentiable at [math]\displaystyle{ t }[/math]^[1] if there exist [math]\displaystyle{ n }[/math] vectors [math]\displaystyle{ e_1, \ldots, e_n }[/math] in [math]\displaystyle{ Y, }[/math] called the partial derivatives of [math]\displaystyle{ f }[/math] at [math]\displaystyle{ t }[/math], such that [math]\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 }[/math] where [math]\displaystyle{ p = \left(p_1, \ldots, p_n\right). }[/math] If [math]\displaystyle{ f }[/math] is differentiable at a point then it is continuous at that point.^[1] If [math]\displaystyle{ f }[/math] is differentiable at every point in some subset [math]\displaystyle{ S }[/math] of its domain then [math]\displaystyle{ f }[/math] is said to be (once or [math]\displaystyle{ 1 }[/math]-time) differentiable in [math]\displaystyle{ S }[/math], where if the subset [math]\displaystyle{ S }[/math] is not mentioned then this means that it is differentiable at every point in its domain. If [math]\displaystyle{ f }[/math] is differentiable and if each of its partial derivatives is a continuous function then [math]\displaystyle{ f }[/math] is said to be (once or [math]\displaystyle{ 1 }[/math]-time) continuously differentiable or [math]\displaystyle{ C^1. }[/math]^[1] For [math]\displaystyle{ k \in \N, }[/math] having defined what it means for a function [math]\displaystyle{ f }[/math] to be [math]\displaystyle{ C^k }[/math] (or [math]\displaystyle{ k }[/math] times continuously differentiable), say that [math]\displaystyle{ f }[/math] is [math]\displaystyle{ k + 1 }[/math] times continuously differentiable or that [math]\displaystyle{ f }[/math] is [math]\displaystyle{ C^{k+1} }[/math] if [math]\displaystyle{ f }[/math] is continuously differentiable and each of its partial derivatives is [math]\displaystyle{ C^k. }[/math] Say that [math]\displaystyle{ f }[/math] is [math]\displaystyle{ C^{\infty}, }[/math] smooth, [math]\displaystyle{ C^\infty, }[/math] or infinitely differentiable if [math]\displaystyle{ f }[/math] is [math]\displaystyle{ C^k }[/math] for all [math]\displaystyle{ k = 0, 1, \ldots. }[/math] The support of a function [math]\displaystyle{ f }[/math] is the closure (taken in its domain [math]\displaystyle{ \operatorname{domain} f }[/math]) of the set [math]\displaystyle{ \{ x \in \operatorname{domain} f : f(x) \neq 0 \}. }[/math]

2. Spaces of C^k 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 [math]\displaystyle{ C^k(\Omega;Y) }[/math] that was constructed could (up to TVS-isomorphism) have instead been defined simply as the completed injective tensor product [math]\displaystyle{ C^k(\Omega) \widehat{\otimes}_{\epsilon} Y }[/math] of the usual space of smooth test functions [math]\displaystyle{ C^k(\Omega) }[/math] with [math]\displaystyle{ Y. }[/math]

Throughout, let [math]\displaystyle{ Y }[/math] be a Hausdorff topological vector space (TVS), let [math]\displaystyle{ k \in \{ 0, 1, \ldots, \infty \}, }[/math] and let [math]\displaystyle{ \Omega }[/math] be either:

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

2.1. Space of C^k Functions

For any [math]\displaystyle{ k = 0, 1, \ldots, \infty, }[/math] let [math]\displaystyle{ C^k(\Omega;Y) }[/math] denote the vector space of all [math]\displaystyle{ C^k }[/math] [math]\displaystyle{ Y }[/math]-valued maps defined on [math]\displaystyle{ \Omega }[/math] and let [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] denote the vector subspace of [math]\displaystyle{ C^k(\Omega;Y) }[/math] consisting of all maps in [math]\displaystyle{ C^k(\Omega;Y) }[/math] that have compact support. Let [math]\displaystyle{ C^k(\Omega) }[/math] denote [math]\displaystyle{ C^k(\Omega;\mathbb{F}) }[/math] and [math]\displaystyle{ C_c^k(\Omega) }[/math] denote [math]\displaystyle{ C_c^k(\Omega; \mathbb{F}). }[/math] Give [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] the topology of uniform convergence of the functions together with their derivatives of order [math]\displaystyle{ \lt k + 1 }[/math] on the compact subsets of [math]\displaystyle{ \Omega. }[/math]^[1] Suppose [math]\displaystyle{ \Omega_1 \subseteq \Omega_2 \subseteq \cdots }[/math] is a sequence of relatively compact open subsets of [math]\displaystyle{ \Omega }[/math] whose union is [math]\displaystyle{ \Omega }[/math] and that satisfy [math]\displaystyle{ \overline{\Omega_i} \subseteq \Omega_{i+1} }[/math] for all [math]\displaystyle{ i. }[/math] Suppose that [math]\displaystyle{ \left(V_\alpha\right)_{\alpha \in A} }[/math] is a basis of neighborhoods of the origin in [math]\displaystyle{ Y. }[/math] Then for any integer [math]\displaystyle{ \ell \lt k + 1, }[/math] the sets: [math]\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\} }[/math] form a basis of neighborhoods of the origin for [math]\displaystyle{ C^k(\Omega;Y) }[/math] as [math]\displaystyle{ i, }[/math] [math]\displaystyle{ \ell, }[/math] and [math]\displaystyle{ \alpha \in A }[/math] vary in all possible ways. If [math]\displaystyle{ \Omega }[/math] is a countable union of compact subsets and [math]\displaystyle{ Y }[/math] is a Fréchet space, then so is [math]\displaystyle{ C^(\Omega;Y). }[/math] Note that [math]\displaystyle{ \mathcal{U}_{i, l, \alpha} }[/math] is convex whenever [math]\displaystyle{ U_{\alpha} }[/math] is convex. If [math]\displaystyle{ Y }[/math] is metrizable (resp. complete, locally convex, Hausdorff) then so is [math]\displaystyle{ C^k(\Omega;Y). }[/math]^[1][2] If [math]\displaystyle{ (p_\alpha)_{\alpha \in A} }[/math] is a basis of continuous seminorms for [math]\displaystyle{ Y }[/math] then a basis of continuous seminorms on [math]\displaystyle{ C^k(\Omega;Y) }[/math] is: [math]\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) }[/math] as [math]\displaystyle{ i, }[/math] [math]\displaystyle{ \ell, }[/math] and [math]\displaystyle{ \alpha \in A }[/math] vary in all possible ways.^[1]

2.2. 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 [math]\displaystyle{ K \subseteq \Omega, }[/math] denote the set of all [math]\displaystyle{ f }[/math] in [math]\displaystyle{ C^k(\Omega;Y) }[/math] whose support lies in [math]\displaystyle{ K }[/math] (in particular, if [math]\displaystyle{ f \in C^k(K;Y) }[/math] then the domain of [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \Omega }[/math] rather than [math]\displaystyle{ K }[/math]) and give it the subspace topology induced by [math]\displaystyle{ C^k(\Omega;Y). }[/math]^[1] If [math]\displaystyle{ K }[/math] is a compact space and [math]\displaystyle{ Y }[/math] is a Banach space, then [math]\displaystyle{ C^0(K;Y) }[/math] becomes a Banach space normed by [math]\displaystyle{ \| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|. }[/math]^[2] Let [math]\displaystyle{ C^k(K) }[/math] denote [math]\displaystyle{ C^k(K;\mathbb{F}). }[/math] For any two compact subsets [math]\displaystyle{ K \subseteq L \subseteq \Omega, }[/math] the inclusion [math]\displaystyle{ \operatorname{In}_{K}^{L} : C^k(K;Y) \to C^k(L;Y) }[/math] is an embedding of TVSs and that the union of all [math]\displaystyle{ C^k(K;Y), }[/math] as [math]\displaystyle{ K }[/math] varies over the compact subsets of [math]\displaystyle{ \Omega, }[/math] is [math]\displaystyle{ C_c^k(\Omega;Y). }[/math]

2.3. Space of Compactly Support C^k Functions

For any compact subset [math]\displaystyle{ K \subseteq \Omega, }[/math] let [math]\displaystyle{ \operatorname{In}_K : C^k(K;Y) \to C_c^k(\Omega;Y) }[/math] denote the inclusion map and endow [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] with the strongest topology making all [math]\displaystyle{ \operatorname{In}_K }[/math] continuous, which is known as the final topology induced by these map. The spaces [math]\displaystyle{ C^k(K;Y) }[/math] and maps [math]\displaystyle{ \operatorname{In}_{K_1}^{K_2} }[/math] form a direct system (directed by the compact subsets of [math]\displaystyle{ \Omega }[/math]) whose limit in the category of TVSs is [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] together with the injections [math]\displaystyle{ \operatorname{In}_{K}. }[/math]^[1] The spaces [math]\displaystyle{ C^k\left(\overline{\Omega_i}; Y\right) }[/math] and maps [math]\displaystyle{ \operatorname{In}_{\overline{\Omega_i}}^{\overline{\Omega_j}} }[/math] also form a direct system (directed by the total order [math]\displaystyle{ \mathbb{N} }[/math]) whose limit in the category of TVSs is [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] together with the injections [math]\displaystyle{ \operatorname{In}_{\overline{\Omega_i}}. }[/math]^[1] Each embedding [math]\displaystyle{ \operatorname{In}_K }[/math] is an embedding of TVSs. A subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] is a neighborhood of the origin in [math]\displaystyle{ C_c^k(\Omega;Y) }[/math] if and only if [math]\displaystyle{ S \cap C^k(K;Y) }[/math] is a neighborhood of the origin in [math]\displaystyle{ C^k(K;Y) }[/math] for every compact [math]\displaystyle{ K \subseteq \Omega. }[/math] This direct limit topology (i.e. the final topology) on [math]\displaystyle{ C_c^\infty(\Omega) }[/math] is known as the canonical LF topology.

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

2.4. Properties

2.5. Identification as a Tensor Product

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

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