Hereditarily indecomposable Banach space

An infinite-dimensional Banach space is hereditarily indecomposable if no closed infinite-dimensional subspace is a topological direct sum of two infinite-dimensional closed subspaces. Equivalently, for any two infinite-dimensional closed subspaces $U,V$ and any $\epsilon >0$, some unit vectors $u\in U,v\in V$ satisfy $\Vert u-v\Vert <\epsilon$. A positive lower bound on such distances would make the addition map from $U\oplus V$ an isomorphism onto a closed subspace.

Every infinite-dimensional Banach space contains either an unconditional basic sequence or a hereditarily indecomposable Banach space. For a space with a basis and no unconditional basic sequence, apply the block Ramsey theorem to finite sequences whose odd and even spans contain nearby unit vectors. A diagonal sequence of successively sharper games gives a subspace in which every pair of infinite-dimensional subspaces has angle zero.