A đâ-saturated Banach space with no long unconditional basic sequences
Bibliographic record
Abstract
We present a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a Schauder basis of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:msub> <mml:mi> Ï </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is saturated by copies of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">c_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and such that for every closed decomposition of a closed subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals upper X 0 circled-plus upper X 1"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> â </mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X=X_0\oplus X_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 0"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">X_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 1"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">X_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have âfew operatorsâ in the sense that every bounded operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper X right-arrow German upper X"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> â </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">T:X \rightarrow \mathfrak {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {X}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sum of a multiple of the inclusion and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 1"> <mml:semantics> <mml:msub> <mml:mi> Ï </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -singular operator, i.e., an operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not an isomorphism on any non-separable subspace of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also show that while <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper X">
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frameâs 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.015 | 0.006 |
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".