Free Banach lattices
Bibliographic record
Abstract
We investigate the structure of the free p -convex Banach lattice {\mathrm{FBL}}^{(p)}[E] over a Banach space E . After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator T:E\rightarrow F between Banach spaces transfer to the associated lattice homomorphism \bar{T}:{\mathrm{FBL}}^{(p)}[E]\rightarrow{\mathrm{FBL}}^{(p)}[F] . Particular consideration is devoted to the case when the operator T is an isomorphic embedding, which leads us to examine extension properties of operators into \ell_{p} , and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is provided. Among other things, this allows us to settle an a priori unrelated question, providing the first instance of a subspace of a Banach lattice without bibasic sequences. In addition, we begin to build a dictionary between Banach space properties of E and Banach lattice properties of {\mathrm{FBL}}^{(p)}[E] . In particular, we characterize the existence of lattice copies of \ell_{1} in {\mathrm{FBL}}^{(p)}[E] and show that \mathrm{FBL}[E] has an upper p -estimate if and only if \textup{id}_{E^*} is (q,1) -summing ( {1}/{p}+{1}/{q}=1 ). We also highlight the significant differences between {\mathrm{FBL}}^{(p)} -spaces depending on whether p is finite or infinite. For example, we show that \mathrm{FBL}^{(\infty)}[E] is lattice isometric to \mathrm{FBL}^{(\infty)}[F] whenever E and F have monotone finite-dimensional decompositions, while, on the other hand, when p<\infty and E^{*} is smooth, {\mathrm{FBL}}^{(p)}[E] determines E isometrically.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 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".