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

TL;DR

This work formulates p-adic analogues of three foundational results—Grothendieck's inequality, Johnson-Lindenstrauss flattening, and Bourgain-Tzafriri restricted invertibility—within non-Archimedean Hilbert spaces. It defines a p-adic inner product and norm (for example, on $\mathbb{Q}_p^d$, $\langle (a_j),(b_j)\rangle=\sum_j a_j b_j$, $\|(x_j)\|=\max_j |x_j|$) and poses precise questions about a universal constant $K_{\mathbb{K}}$, optimal dimension bounds $m$, and invertibility constants in the p-adic setting. The resulting framework provides a foundational step toward p-adic functional analysis and potential applications in non-Archimedean harmonic analysis. Overall, the paper lays out concrete p-adic analogues that can guide future theoretical development and cross-domain research.

Abstract

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

Theorems & Definitions (8)

• Theorem 1.1
• Definition 1.2
• Example 1.3
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1