Complemented Subspaces of Linear Bounded Operators
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Bibliographic record
Abstract
Abstract We study the complementation of the space W ( X , Y ) of weakly compact operators, the space K ( X , Y ) of compact operators, the space U ( X , Y ) of unconditionally converging operators, and the space CC ( X , Y ) of completely continuous operators in the space L ( X , Y ) of bounded linear operators from X to Y . Feder proved that if X is infinite-dimensional and c 0 ↪ Y , then K ( X , Y ) is uncomplemented in L ( X , Y ). Emmanuele and John showed that if c 0 ↪ K ( X , Y ), then K ( X , Y ) is uncomplemented in L ( X , Y ). Bator and Lewis showed that if X is not a Grothendieck space and c 0 ↪ Y , then W ( X , Y ) is uncomplemented in L ( X , Y ). In this paper, classical results of Kalton and separably determined operator ideals with property (∗) are used to obtain complementation results that yield these theorems as corollaries.
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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) | 0.039 | 0.003 |
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 it