Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 11 / 2 / a / Solution

An infinite-dimensional Banach space $X$ is a hereditarily indecomposable Banach space when no closed infinite-dimensional subspace of $X$ splits as a topological direct sum of two infinite-dimensional closed subspaces. Finite-dimensional summands are not forbidden.

A useful equivalent formulation is that for every pair of infinite-dimensional closed subspaces $U,V$ and every $\epsilon >0$, there are unit vectors $u\in U,v\in V$ with $\Vert u-v\Vert <\epsilon$. To see the equivalence, a topological direct sum has bounded coordinate projections and hence a positive separation between its two unit spheres. Conversely, if the two unit spheres have separation $\delta >0$, then $U\cap V=0$ and addition is bounded below on $U\oplus V$: after factoring out the larger of $\Vert u\Vert ,\Vert v\Vert$, the triangle inequality and the separation bound give a positive lower bound for $\Vert u+v\Vert /(\Vert u\Vert +\Vert v\Vert )$. Its range is therefore closed and supplies the forbidden decomposition.