The Space of Test Functions

We construct the space $\mathcal{D}(\Omega)$ and equip it with the topology that determines which linear functionals qualify as distributions.

Definition¶

Throughout, $\Omega \subseteq \mathbb{R}^d$ is an open set. We use standard multi-index notation: for $\alpha = (\alpha_1, \ldots, \alpha_d) \in \mathbb{N}_0^d$, we write $|\alpha| = \alpha_1 + \cdots + \alpha_d$ and $D^\alpha = \partial_1^{\alpha_1} \cdots \partial_d^{\alpha_d}$.

Test functions are the “probes” we use to measure distributions. They are infinitely differentiable and vanish outside a bounded region, so integrals against them are always well-defined.

Notation. In the duality chapter, we wrote $X'$ for the algebraic dual (all linear functionals) and $X^*$ for the topological dual (continuous linear functionals). The standard convention in distribution theory, due to Schwartz, writes $\mathcal{D}'(\Omega)$ for the space of distributions, which is the topological dual of $\mathcal{D}(\Omega)$. This conflicts with our earlier notation, but is universal in the literature. Throughout this chapter, $\mathcal{D}'(\Omega)$ always means the topological dual, i.e. the space of continuous linear functionals on $\mathcal{D}(\Omega)$.

Why $C_c^\infty$ is rich enough¶

Before diving into the topology of $\mathcal{D}(\Omega)$, we address the first natural question: is this space large enough to be useful?

$\mathcal{D}(\Omega)$ is dense in $L^p(\Omega)$ for $1 \leq p < \infty$ and in $C_0(\Omega)$. This follows from the approximation results in the function spaces chapter: continuous compactly supported functions are dense in $L^p$, and mollification promotes them to smooth compactly supported functions.

This density is a special case of a general principle.

The space $\mathcal{D}(\Omega)$ satisfies both hypotheses on $X = \Omega$: smooth bump functions can separate any two points, and for every $x \in \Omega$ one can construct a bump with $\varphi(x) > 0$. Stone-Weierstrass therefore gives $\mathcal{D}(\Omega)$ dense in $C_0(\Omega)$, and since $C_0$ is dense in $L^p$ for $1 \leq p < \infty$, density in $L^p$ follows by transitivity.

$\mathcal{D}(\Omega)$ is also dense in the Sobolev spaces $W^{k,p}(\Omega)$ for $1 \leq p < \infty$ (the Meyers-Serrin theorem, $H = W$). This is a deeper result that does not follow from Stone-Weierstrass alone — it requires the calculus of mollification to commute with weak derivatives.

The inductive limit topology¶

The solution is to build the topology in two stages Tao, 2009.

Local convexity. A topological vector space is locally convex if every point has a neighborhood basis consisting of convex sets, or equivalently, if the topology is generated by a family of seminorms. Each $\mathcal{D}_{K_j}$ is locally convex because its topology comes from the seminorms $\|\cdot\|_{K_j,k}$. The inductive limit topology on $\mathcal{D}(\Omega)$ is locally convex by construction: it is defined as the finest locally convex topology making the inclusions $\iota_j$ continuous (Definition 4). Local convexity matters because it is the hypothesis of the geometric Hahn-Banach theorem (the separation theorem). Without it, the dual space $\mathcal{D}'(\Omega)$ might be too small to separate points of $\mathcal{D}(\Omega)$.

Why not a single norm? In a Banach space, the topology is determined by a single norm. In a Fréchet space, it comes from countably many seminorms. For $\mathcal{D}(\Omega)$, neither works: the topology must enforce both that all derivatives converge uniformly and that supports stay bounded. No single norm (nor even countably many seminorms) can do both. Each $\mathcal{D}_K$ is Fréchet, with seminorms $\|\cdot\|_{K,k}$, but $\mathcal{D}(\Omega) = \bigcup_K \mathcal{D}_K$ is a union over all compact $K \subset \Omega$, and no single compact set suffices. The inductive limit topology is precisely the tool that handles this union.

This is not an abstract curiosity; it has a clean sequential characterization.

This is a very strong notion of convergence. Both conditions must hold: derivatives of all orders must converge uniformly, and the supports cannot wander off to infinity. For comparison, convergence in $L^2$ only requires $\int |\varphi_n - \varphi|^2 \to 0$, which says nothing about derivatives and nothing about where the mass lives.

Not metrizable, but sequences suffice. A topological space is first countable if every point has a countable neighborhood basis (a sequence $U_1 \supset U_2 \supset \cdots$ such that every neighborhood contains some $U_n$). Every metrizable space is first countable (take $U_n = B(x, 1/n)$). The space $\mathcal{D}(\Omega)$ is not first countable: a neighborhood of 0 must restrict to a neighborhood of 0 in every $\mathcal{D}_{K_j}$, and the “size” of these restrictions can be chosen independently, so no countable collection suffices. In general, sequential continuity does not imply continuity in spaces that are not first countable. However, the inductive limit structure gives a remarkable exception: for linear maps $T : \mathcal{D}(\Omega) \to X$ into a locally convex space, continuity is equivalent to sequential continuity. This is because $T$ is continuous if and only if each restriction $T \circ \iota_j : \mathcal{D}_{K_j} \to X$ is continuous, and each $\mathcal{D}_{K_j}$ is metrizable. This is why we can work entirely with sequences throughout distribution theory.

Characterizing continuous functionals¶

The inductive limit topology determines exactly which linear functionals on $\mathcal{D}(\Omega)$ are continuous.

What transfers from earlier chapters?¶

The spaces $\mathcal{D}_K$, $\mathcal{D}(\Omega)$, and $\mathcal{D}'(\Omega)$ are progressively further from Banach spaces:

The key theorems from the linear operators chapter were proved using the Baire category theorem, which requires completeness and metrizability. Each $\mathcal{D}_K$ has both, so the theorems apply there. The full space $\mathcal{D}(\Omega)$ is not metrizable, so we cannot apply them directly, but we can apply them on each piece $\mathcal{D}_K$ and glue the results together.

The recurring pattern: do Fréchet space arguments on each $\mathcal{D}_K$, then glue via the inductive limit. The inductive limit structure is the price we pay for the compact support condition; the reward is a dual space ($\mathcal{D}'$) where differentiation always makes sense.