An elementary and unified proof of Grothendieck's inequality

TL;DR

This paper provides an elementary, unified proof of Grothendieck's inequality that simplifies existing proofs and includes the best-known bounds for real and complex cases, making it pedagogically accessible.

Contribution

It offers a self-contained, unified proof of Grothendieck's inequality that incorporates the Krivine and Haagerup bounds, streamlining previous complex proofs.

Findings

Unified proof for real and complex Grothendieck's inequality

Derivation of Krivine and Haagerup bounds within the proof

Elementary approach requiring basic calculus and linear algebra

Abstract

We present an elementary, self-contained proof of Grothendieck's inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current best-known explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss--Pe{\l}czy\'nski, Krivine, and Haagerup into a proof that need only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.