Result 326, Functional analysis

The cotype–cotype conjecture under the approximation property

Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.

Lean formalization Proof

The bigger picture

Why it matters

Randomly adding and subtracting vectors can reveal a space's geometry. The manuscript identifies when constraints on a space and its linear measurements together guarantee stronger geometric regularity, under a specified approximation assumption.

What changes?

The manuscript reports that a nonzero real Banach space, a complete normed vector space, with the ordinary approximation property is K-convex if and only if both it and its dual have finite Rademacher cotype, possibly with different exponents. The approximation property means that finite-rank continuous linear maps approximate the identity uniformly on compact sets. The dual consists of continuous linear measurements of vectors. Here K-convexity is equivalent to nontrivial Rademacher type.

What does that help mathematicians do?

Cotype limits how small random signed sums can be on average relative to their summands' sizes. Nontrivial type provides upper control stronger than the triangle inequality. The claimed equivalence therefore turns information about the space and its dual into control in the opposite direction. Conversely, within the stated class, failure of K-convexity forces at least one of the two spaces to lack finite cotype, ruling out otherwise conceivable combinations of geometric properties.

Are there practical applications?

Its immediate value is foundational: researchers could use cotype information about a space and its dual to establish nontrivial type without estimating it directly. This links two ways of studying random vector sums. The supplied statement does not specify quantitative type bounds or cover spaces without the approximation property.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The cotype–cotype conjecture under the approximation property

September 23, 2026 21 pages Main result formalized in Lean

We prove the cotype–cotype conjecture under the ordinary approximation property. A nonzero real Banach space with this property is K-convex if and only if both the space and its dual have finite Rademacher cotype, possibly with different exponents.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026,
  author = {{OpenAI}},
  title = {{The cotype--cotype conjecture under the approximation property}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026/paper.pdf}{OAI:The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/326.md.

The cotype–cotype conjecture under the approximation property

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The cotype–cotype problem asks whether finite cotype of a Banach space and its dual characterizes KK-convexity. The formalized result establishes this equivalence for every nonzero real Banach space with the approximation property: XX is KK-convex exactly when both XX and X∗X^* have finite cotype.

Comparator links

Result Comparator statement
Cotype–cotype equivalence with approximation property Cotype.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.