LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Test function inclusion in schwartz space is continuous

Statement

For $n \geq 1$ , the inclusion $ι : D (R^{n}) ↪ S (R^{n})$ is continuous and has dense image. This holds in ZF.

Facts & Assumptions

Given: The LF test space $D (R^{n})$ and Schwartz space $S (R^{n})$ .

[F1]

A linear map from $D$ is continuous exactly when every restriction to $D_{K}$ is continuous (Test function lf topology universal property).

[F2]

Schwartz space is defined by the seminorms $p_{α β}$ and their locally convex topology (Schwartz space and its seminorms, Schwartz topology and convergence).

[F3]

Smooth compactly supported cutoffs approximate every Schwartz function in all Schwartz seminorms (Smooth compact supports are dense in Schwartz space).

Proof

technique · fixed-support estimates and cutoff density

1.1

Fix compact $K \subseteq R^{n}$ and $φ \in D_{K}$ . Each Schwartz seminorm has the following fixed-support bound.

$p_{α β} (φ) \leq (\sup_{x \in K} ∣ x^{α} ∣) \sup_{x \in K} ∣ \partial^{β} φ (x) ∣ .$

The multiplier on the right is finite, including the zero value when $K = \emptyset$ . Hence every Schwartz seminorm pulls back continuously to $D_{K}$ . [F2]

2.1

Step 1.1 makes every restricted inclusion $D_{K} \to S$ continuous. The LF universal property therefore makes $ι$ continuous on all of $D$ .

F1step 1.1

3.1

The approximants in [F3] lie in $D$ and converge in the Schwartz topology to the prescribed Schwartz function. Thus the image of $ι$ is dense. This density argument is separate from continuity, and neither uses a choice axiom.

F3∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)
Radu Gelca, Functional Analysis (standard reference, not scraped)