A simple, combinatorial proof of Gowers Dichotomy for Banach Spaces

TL;DR

This paper provides a straightforward combinatorial proof of Gowers Dichotomy, showing that every infinite-dimensional Banach space contains a subspace with either an unconditional basis or is hereditarily indecomposable, using infinite Ramsey theory.

Contribution

It introduces a simple, combinatorial proof of Gowers Dichotomy, simplifying previous complex proofs and leveraging classical infinite Ramsey theory techniques.

Findings

Every infinite-dimensional Banach space has a subspace with an unconditional basic sequence.

Alternatively, such a space can be hereditarily indecomposable.

The proof is based on combinatorial methods from infinite Ramsey theory.

Abstract

In this paper we present a simple proof of Gowers Dichotomy which states that every infinite dimensional Banach Space has a subspace which either contains an unconditional basic sequence or is hereditarily indecomposable. Our approach is purely combinatorial and mainly based on work of Ellentuck, Galvin and Prikry in infinite Ramsey theory.