= 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 $\varepsilon>0$, there are unit vectors $u\in U,v\in V$ with $\|u-v\|<\varepsilon$. 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 $\|u\|,\|v\|$, the triangle inequality and the separation bound give a positive lower bound for $\|u+v\|/(\|u\|+\|v\|)$. Its range is therefore closed and supplies the forbidden decomposition.
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