A Gleason–Kahane–Żelazko theorem for reproducing kernel Hilbert spaces
Bibliographic record
Abstract
We establish the following Hilbert-space analog of the Gleason–Kahane–Żelazko theorem. If H ${\mathcal {H}}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if Λ $\Lambda$ is a linear functional on H ${\mathcal {H}}$ such that Λ ( 1 ) = 1 $\Lambda (1)=1$ and Λ ( f ) ≠ 0 $\Lambda (f)\ne 0$ for all cyclic functions f ∈ H $f\in {\mathcal {H}}$ , then Λ $\Lambda$ is multiplicative, in the sense that Λ ( f g ) = Λ ( f ) Λ ( g ) $\Lambda (fg)=\Lambda (f)\Lambda (g)$ for all f , g ∈ H $f,g\in {\mathcal {H}}$ such that f g ∈ H $fg\in {\mathcal {H}}$ . Moreover Λ $\Lambda$ is automatically continuous. We give examples to show that the theorem fails if the hypothesis of a complete Pick kernel is omitted. We also discuss conditions under which Λ $\Lambda$ has to be a point evaluation.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".