Non-complemented Spaces of Operators, Vector Measures, and<i>c<sub>o</sub></i>
Bibliographic record
Abstract
Abstract The Banach spaces L ( X , Y ), K ( X , Y ), L w* ( X *, Y ), and K w* ( X *, Y ) are studied to determine when they contain the classical Banach spaces c o or ℓ ∞ . The complementation of the Banach space K ( X , Y ) in L ( X , Y ) is discussed as well as what impact this complementation has on the embedding of c o or ℓ ∞ in K ( X , Y ) or L ( X , Y ). Results of Kalton, Feder, and Emmanuele concerning the complementation of K ( X , Y ) in L ( X , Y ) are generalized. Results concerning the complementation of the Banach space K w* ( X *, Y ) in L w* ( X *, Y ) are also explored as well as how that complementation affects the embedding of c o or ℓ ∞ in K w* ( X *, Y ) or L w* ( X *, Y ). The ℓ p spaces for 1 = p < ∞ are studied to determine when the space of compact operators from one ℓ p space to another contains c o . The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.
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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 | 0.001 |
| 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) | 0.003 | 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; 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".