01 · The idea
In everyday language
A Banach space is a vector space with a complete notion of length; its vectors may encode functions rather than ordinary arrows. Finite-dimensional intuition suggests choosing coordinates and splitting a space into simpler pieces. Gowers showed how dramatically this intuition can fail. His dichotomy finds either a subspace with an unconditional basis, whose convergent expansions tolerate changes of signs, or a hereditarily indecomposable subspace, none of whose infinite-dimensional subspaces splits into two infinite-dimensional closed pieces. Combinatorial selection arguments uncover this alternative. The conclusion organises possible subspaces rather than declaring all Banach spaces equivalent or easily classifiable.
Why it matters
The work solved central problems in Banach-space theory and made combinatorial methods indispensable to understanding infinite-dimensional geometry, including spaces with unexpectedly few possible decompositions or symmetries.
