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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.

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.

A bounded linear operator is strictly singular if it is bounded below on no infinite-dimensional subspace. Equivalently, every infinite-dimensional subspace contains unit vectors with arbitrarily small images. On a hereditarily indecomposable Banach space, an operator which is arbitrarily small on unit vectors in every finite-codimensional subspace is strictly singular: construct an infinite basic span on which its operator norm is small, then use the zero-angle property of hereditary indecomposability.