p-adic Grothendieck Inequality, p-adic Johnson-Lindenstrauss Flattening and p-adic Bourgain-Tzafriri Restricted Invertibility Problems

We formulate p-adic versions of following three: (1) Grothendieck Inequality, (2) Johnson-Lindenstrauss Flattening Lemma, (3) Bourgain-Tzafriri Restricted Invertibility Theorem.

💡 Research Summary

The manuscript sets out to formulate p‑adic analogues of three cornerstone results in functional analysis and high‑dimensional geometry: Grothendieck’s inequality, the Johnson‑Lindenstrauss (JL) flattening lemma, and the Bourgain‑Tzafriri (BT) restricted invertibility theorem. After recalling the definition of a p‑adic Hilbert space—i.e., a vector space over a non‑Archimedean valued field K equipped with a bilinear, symmetric inner product satisfying the ultrametric Cauchy‑Schwarz inequality—the author introduces three open problems that ask whether the essential quantitative features of the classical theorems survive in the non‑Archimedean setting.

p‑adic Grothendieck problem.
Problem 1.4 asks whether there exists a universal constant (K_{K}) such that for every p‑adic Hilbert space X, any scalar matrix (A=