Definition and Topology#

The space ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ consists of all infinitely differentiable functions with compact support on ${\mathbb{R}}^d$.

To define the topology on ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$:

1. View ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ as the union of spaces ${\mathcal{C}}_c^{\mathrm{\infty }}(K)$ where $K\subset {\mathbb{R}}^d$ is compact.

2. We equip each ${\mathcal{C}}_c^{\mathrm{\infty }}(K)$ with the smooth topology generated by the norm \(||f||_{\mathcal{C}^k\).

3. Define seminorms as good when they are continuous on each ${\mathcal{C}}_c^{\mathrm{\infty }}(K)$.

4. Using these good seminorms we generate the topology on ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$.

This gives ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ the structure of a locally convex topological vector space.

Convergence in the Space of Test Functions#

A sequence $\left\{f_n\right\}$ in ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ converges to $f$ if and only if:

1. There exists a compact set $K$ such that all $f_n$ and $f$ are supported in $K$.

2. For every multi-index $\alpha$, ${\partial }^{\alpha }{f}_n$ converges uniformly to ${\partial }^{\alpha }f$.

This is a very strong notion of convergence, requiring uniform convergence of all derivatives and eventual containment of supports in a fixed compact set.

Properties of test functions#

1. Density: ${\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ is dense in many function spaces including $L^p$ for $1\le p<\mathrm{\infty }$ and ${\mathcal{C}}_0$.

2. Partitions of Unity: For any open cover of a compact set $K$, there exists a partition of unity consiting of test functions.

3. Convolution: If $f\in {\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ and $g$ locally integrable and compactly supported, then $f\ast g\in {\mathcal{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^d)$.

4. Approximation to Identity: There exist sequences of test functions that converge to the Dirac delta distribution.