Non-complemented Spaces of Operators, Vector Measures, and<i>c<sub>o</sub></i>
Bibliographic record
Abstract
Abstract The Banach spacesL(X,Y),K(X,Y),Lw*(X*,Y), andKw*(X*,Y) are studied to determine when they contain the classical Banach spacescoor ℓ∞. The complementation of the Banach spaceK(X,Y) inL(X,Y) is discussed as well as what impact this complementation has on the embedding ofcoor ℓ∞inK(X,Y) orL(X,Y). Results of Kalton, Feder, and Emmanuele concerning the complementation ofK(X,Y) inL(X,Y) are generalized. Results concerning the complementation of the Banach spaceKw*(X*,Y) inLw*(X*,Y) are also explored as well as how that complementation affects the embedding ofcoor ℓ∞inKw*(X*,Y) orLw*(X*,Y). The ℓpspaces for 1 =p< ∞ are studied to determine when the space of compact operators from one ℓpspace to another containsco. 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 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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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".