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Math-2698 · mathematics · Analysis

The cotype-cotype conjecture with the approximation property

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every nonzero real Banach space X with the ordinary approximation property, is X K-convex if and only if both X and its continuous dual have finite cotype, allowing different finite exponents for the two spaces?

Short English statement

For every nonzero real Banach space X with the ordinary approximation property, is X K-convex if and only if both X and its continuous dual have finite cotype, allowing different finite exponents for the two spaces?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2698-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2698

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 326; 2026-09-23).

The cotype-cotype conjecture with the approximation property

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 326; 2026-09-23), the asserted result is that when the ordinary approximation property holds, a nonzero real Banach space $X$ is $K$-convex precisely when $X$ and its dual each have some finite cotype.

Mathematical statement of the claimed result

Manuscript 1.
Take a real Banach space $X\ne\{0\}$ and assume ordinary AP: finite-rank bounded operators approximate the identity uniformly on compact sets, without a common norm bound. Define Rademacher cotype by \[ (\sum_i\|x_i\|^q)^{1/q}\le C_q(X)(\mathbb E\|\sum_i\varepsilon_ix_i\|^2)^{1/2}\quad(q\ge2), \] and $K$-convexity by uniform boundedness, on vector-valued $L^2$ sign cubes, of the first-order Rademacher projections. The claim is \[ X\text{ is }K\text{-convex}\ \Longleftrightarrow\ (\exists q,r\in[2,\infty):C_q(X)<\infty,\ C_r(X^*)<\infty). \] Exponents may differ. The forward implication needs no AP, while the reverse implication retains it.

Outline of the argument

For manuscript 1 (family 326; 2026-09-23), walsh moment lifts factor finite-dimensional data through a compact domain with controlled dual moments. Approximation by finite-rank maps then gives uniform Walsh decay, and a scalar coefficient estimate excludes the dimension growth of a non-$K$-convex space.

Formal verification

For manuscript 1 (family 326; 2026-09-23), OpenAI lists Lean statements for this assertion: when the ordinary approximation property holds, a nonzero real Banach space $X$ is $K$-convex precisely when $X$ and its dual each have some finite cotype.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are Cotype.lean.

Their named declarations include OAI.Cotype.mainTarget_proved.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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