Differentiable vector-valued functions from Euclidean space: Difference between revisions

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In this section, the definition of the canonical LF-topology on the [[space of smooth test functions]], and the topologies needed for its definition, is generalized to functions valued in general TVSs.

Throughout, let <math>k \in \{ 0, 1, \ldots, \infty \}</math> and let <math>\Omega</math> be either:

# an open subset of <math>\R^n,</math> where <math>n \geq 1</math> is an integer, or else

# a [[locally compact]] topological space, in which case <math>k</math> can only be <math>0,</math> and let <math>Y</math> be a [[topological vector space]] (TVS).

=== Space of ''C''^''k'' functions ===

For any <math>k = 0, 1, \ldots, \infty,</math> let <math>C^k(\Omega;Y)</math> denote the vector space of all <math>C^k</math> <math>Y</math>-valued maps defined on <math>\Omega</math> and let <math>C_c^k(\Omega;Y)</math> denote the vector subspace of <math>C^k\left( \Omega; Y \right)</math> consisting of all maps in <math>C^k(\Omega;Y)</math> that have compact support.

Let <math>C^k(\Omega)</math> denote <math>C^k\left( \Omega; \mathbb{F} \right)</math> and <math>C_c^k\left( \Omega \right)</math> denote <math>C_c^k(\Omega; \mathbb{F}).</math>

Give <math>C_c^k(\Omega;Y)</math> the topology of uniform convergence of the functions together with their derivatives of order <math>< k + 1</math> on the compact subsets of <math>\Omega.</math>{{sfn|Trèves|2006|pp=412–419}}

Suppose <math>\Omega_1 \subseteq \Omega_2 \subseteq \cdots</math> is a sequence of [[relatively compact]] open subsets of <math>\Omega</math> whose union is <math>\Omega</math> and that satisfy <math>\overline{\Omega_i} \subseteq \Omega_{i+1}</math> for all <math>i.</math>

Suppose that <math>\left(V_\alpha\right)_{\alpha \in A}</math> is a basis of neighborhoods of the origin in <math>Y.</math> Then for any integer <math>\ell < k + 1,</math> the sets:

:<math>\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>C^k(\Omega;Y)</math> as <math>i,</math> <math>l\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.

If <math>\Omega</math> is a countable union of compact subsets and <math>Y</math> is a [[Fréchet space]], then so is <math>C^(\Omega;Y).</math>

Note that <math>\mathcal{U}_{i, l, \alpha}</math> is convex whenever <math>U_{\alpha}</math> is convex.

If <math>Y</math> is [[Metrizable topological vector space|metrizable]] (resp. [[Complete topological vector space|complete]], [[Locally convex topological vector space|locally convex]], [[Hausdorff space|Hausdorff]]) then so is <math>C^k\left( \Omega; Y \right).</math>{{sfn|Trèves|2006|pp=412–419}}{{sfn|Trèves|2006|pp=446–451}}

If <math>(p_\alpha)_{\alpha \in A}</math> is a basis of continuous seminorms for <math>Y</math> then a basis of continuous seminorms on <math>C^k(\Omega;Y)</math> is:

:<math>\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>i,</math> <math>l\ell,</math> and <math>\alpha \in A</math> vary in all possible ways.{{sfn|Trèves|2006|pp=412–419}}

If <math>\Omega</math> is a compact space and <math>Y</math> is a Banach space, then <math>C^0\left( \Omega; Y \right)</math> becomes a Banach space normed by <math>\| f \| := \sup_{\omega \in \Omega} \| f(\omega) \|.</math>{{sfn|Trèves|2006|pp=446–451}}

=== 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>K \subseteq \Omega,</math> let <math>C^k(K; Y\right)</math> denote the set of all <math>f</math> in <math>C^k\left( \Omega; Y \right)</math> whose support lies in <math>K</math> (in particular, if <math>f \in C^k(K; Y)</math> then the ___domain of <math>f</math> is <math>\Omega</math> rather than <math>K</math>) and give <math>C^k(K; Y\right)</math> the subspace topology induced by <math>C^k\left( \Omega; Y \right).</math>{{sfn|Trèves|2006|pp=412–419}}

Let <math>C^k(K)</math> denote <math>C^k\left( K; \mathbb{F} \right).</math>

Note that for any two compact subsets <math>K_1K \subseteq K_2L \subseteq \Omega,</math> the natural inclusion <math>\operatorname{In}_{K_1K}^{K_2L} : C^k\left( K_1K; Y \right) \to C^k\left( K_2L; Y \right)</math> is an embedding of TVSs and that the union of all <math>C^k(K; Y),</math> as <math>K</math> varies over the compact subsets of <math>\Omega,</math> is <math>C_c^k\left( \Omega; Y \right).</math>

=== Space of compactly support ''C''^''k'' functions ===

For any compact subset <math>K \subseteq \Omega,</math> let <math>\operatorname{In}_K : C^k(K; Y) \to C_c^k(\Omega; Y)</math> be the natural inclusion and give <math>C_c^k(\Omega; Y)</math> the strongest topology making all <math>\operatorname{In}_K</math> continuous.

The spaces <math>C^k(K;Y)</math> and maps <math>\operatorname{In}_{K_1}^{K_2}</math> form a [[direct limit|direct system]] (directed by the compact subsets of <math>\Omega</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the natural injections <math>\operatorname{In}_{K}.</math>{{sfn|Trèves|2006|pp=412–419}}

The spaces <math>C^k\left( \overline{\Omega_i}; Y \right)</math> and maps <math>\operatorname{In}_{\overline{\Omega_i}}^{\overline{\Omega_j}}</math> also form a [[direct limit|direct system]] (directed by the total order <math>\mathbb{N}</math>) whose limit in the category of TVSs is <math>C_c^k(\Omega;Y)</math> together with the natural injections <math>\operatorname{In}_{\overline{\Omega_i}}.</math>{{sfn|Trèves|2006|pp=412–419}}

Each natural embedding <math>\operatorname{In}_K</math> is an embedding of TVSs.

A subset <math>S</math> of <math>C_c^k(\Omega;Y)</math> is a neighborhood of the origin in <math>C_c^k(\Omega;Y)</math> if and only if <math>S \cap C^k(K;Y)</math> is a neighborhood of the origin in <math>C^k\left( K; Y \right)</math> for every compact <math>K \subseteq \Omega.</math>

This direct limit topology on <math>C_c^\infty(\Omega)</math> is known as the '''{{em|canonical LF topology}}'''.

If <math>Y</math> is a Hausdorff locally convex space, <math>T</math> is a TVS, and <math>u : C_c^k(\Omega;Y) \to T</math> is a linear map, then <math>u</math> is continuous if and only if for all compact <math>K \subseteq \Omega,</math> the restriction of <math>u</math> to <math>C^k(K;Y)</math> is continuous.{{sfn|Trèves|2006|pp=412–419}} One replace "all compact <math>K \subseteq \Omega</math>" with "all <math>K := \overline{\Omega}_i</math>".

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and let <math>I_k(\phi) : \Delta \to C^k(\Omega)</math> be defined by <math>I_k(\phi)(y) := \phi_y.</math>

Then <math>I_\infty : C^\infty(\Omega \times \Delta) \to C^\infty(\Delta; C^\infty(\Omega))</math> is a (surjective) isomorphism of TVSs.

Furthermore, the restriction <math>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)</math> is an isomorphism of TVSs when <math>C_c^\infty\left( \Omega \times \Delta \right)</math> has its canonical LF topology.

}}

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=412-419}}|note=|style=|math_statement=

Let <math>Y</math> be a Hausdorff locally convex space.

For every continuous linear form <math>y^{\prime} \in Y</math> and every <math>f \in C^\infty(\Omega; Y),</math> let <math>J_{y^{\prime}}(f) : \Omega \to \mathbb{F}</math> be defined by <math>J_{y^{\prime}}(f)(p) = y^{\prime}(f(p)).</math>

Then <math>J_{y^{\prime}} : C^\infty(\Omega; Y) \to C^\infty(\Omega)</math> is a continuous linear map;

and furthermore, the restriction <math>J_{y^{\prime}}\big\vert_{C_c^\infty( \Omega; Y)} : C_c^\infty(\Omega; Y) \to C^\infty(\Omega)</math> is also continuous (where <math>C_c^\infty(\Omega;Y)</math> has the canonical LF topology).

}}