Uniform bounds and moving distributional tests

Selected from AN-01, Order, positivity and distributional limits, Theorem 5.1. The reconstructed source is by GPT-6 Astra (OpenAI), Ultra, October 2026; its earlier edition is by GPT-6.1 Sol (OpenAI), Ultra. This selected CC0 exposition preserves the common-order and moving-test proof. Its stronger finite-net conclusion is not needed or imported. The sole adjustment is to state the common compact support explicitly, so no global test-topology theorem is hidden.

Let X be open in Euclidean space. A distribution is a complex-linear functional on smooth compactly supported functions such that on each fixed compact K it obeys a bound C p(K,m), where p(K,m) is the maximum of the supremum norms of all derivatives through order m. Write D_K for that fixed-support space. Weak convergence means convergence on every fixed smooth compact test.

Theorem U1 (weak completeness and uniform local order). Suppose $u_j\in\mathcal D'(X)$ and $u_j(\phi)$ converges for each smooth compact test. The limits define a distribution $u$. For every fixed compact $K$, one constant $C_K$ and one order $m_K$ work in the local finite-order estimate stated above for all $u_j$ and for $u$.

If the tests $\phi_j,\phi$ have one common compact support and $\phi_j\to\phi$ uniformly with every derivative, then $u_j(\phi_j)\to u(\phi)$.

Proof. Scalar limits preserve linearity. On a fixed $\mathcal D_K$, complete-test-spaces.md, Sections 14.1–14.2, proves that the increasing derivative seminorms give a complete metric topology. The proof takes uniform limits of every derivative of the zero extensions and uses the fundamental theorem on coordinate segments to identify successive derivatives; no regularity of the boundary of $K$ is needed.

For $r=1,2,\ldots$, put

Each is closed, because every $u_j$ is continuous there, and their union is $\mathcal D_K$, because each convergent scalar sequence is bounded. The proved Baire theorem gives a point $\phi_0$ and a neighborhood $p_{K,m}(h)<\varepsilon$ with $\phi_0+h\in A_r$, including $h=0$. Subtraction gives $\sup_j|u_j(h)|\le2r$ in that neighborhood. If $p_{K,m}(h)>0$, apply this to $\varepsilon h/(2p_{K,m}(h))$, obtaining the common bound $4r p_{K,m}(h)/\varepsilon$. If the seminorm is zero, $h=0$, since the zeroth derivative is included. Passing to the scalar limit gives the same bound for $u$, which proves it is a distribution.

For moving tests, their supports lie in one compact by the explicit common-support hypothesis, and

The common derivative bound controls the first term, and fixed-test convergence controls the second. $\square$

Free comparison: Semyon Dyatlov, 18.155 notes, Theorems 4.14 and 4.16, Propositions 4.17–4.18. Their complete used proof is supplied here together with the separately licensed complete-test-space component.