The Space of Test Functions¶

We begin by constructing a suitable space of ``test functions’’ against which distributions will be evaluated.

Definition and Topology¶

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

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

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

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

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

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

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

Convergence in the Space of Test Functions¶

A sequence $\left\{ f_n \right\}$ in $\mathcal{C}_c^\infty(\mathbb{R}^d)$ 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: $\mathcal{C}_c^\infty(\mathbb{R}^d)$ is dense in many function spaces including $L^p$ for $1 \leq p < \infty$ 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 $f \in \mathcal{C}_c^\infty(\mathbb{R}^d)$ and $g$ locally integrable and compactly supported, then $f * g \in \mathcal{C}_c^\infty(\mathbb{R}^d)$.

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