Bibliographic record
Abstract
Abstract Let X and Y be Banach spaces isomorphic to complemented subspaces of each other with supplements A and B . In 1996, W. T. Gowers solved the Schroeder–Bernstein (or Cantor–Bernstein) problem for Banach spaces by showing that X is not necessarily isomorphic to Y . In this paper, we obtain a necessary and sufficient condition on the sextuples ( p , q , r , s , u , v ) in ℕ with p + q ≥ 1, r + s ≥ 1 and u , v ∈ ℕ*, to provide that X is isomorphic to Y , whenever these spaces satisfy the following decomposition scheme Namely, Φ = ( p – u )( s – v )–( q + u )( r + v ) is different from zero and Φ divides p + q and r + s . These sextuples are called Cantor–Bernstein sextuples for Banach spaces. The simplest case (1, 0, 0, 1, 1, 1) indicates the well-known Pełczyński's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder– Bernstein problem become evident.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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; both teacher heads agree on what is shown here.
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".