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 \(\mathcal{C}_c^\infty(\mathbb{R}^d)\):
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.
We equip each \(\mathcal{C}_c^\infty(K)\) with the smooth topology generated by the norm \(||f||_{\mathcal{C}^k\).
Define seminorms as good when they are continuous on each \(\mathcal{C}_c^\infty(K)\).
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:
There exists a compact set \(K\) such that all \(f_n\) and \(f\) are supported in \(K\).
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#
Density: \(\mathcal{C}_c^\infty(\mathbb{R}^d)\) is dense in many function spaces including \(L^p\) for \(1 \leq p < \infty\) and \(\mathcal{C}_0\).
Partitions of Unity: For any open cover of a compact set \(K\), there exists a partition of unity consiting of test functions.
Convolution: If \(f \in \mathcal{C}_c^\infty(\mathbb{R}^d)\) and \(g\) locally integrable and compactly supported, then \(f * g \in \mathcal{C}_c^\infty(\mathbb{R}^d)\).
Approximation to Identity: There exist sequences of test functions that converge to the Dirac delta distribution.