  The theory of low-rank tensor-train approximation is well-understood when the
approximation error is measured in the Frobenius norm. The entrywise maximum
norm is equally important but is significantly weaker for large tensors, making
the estimates obtained via the Frobenius norm and norm equivalence pessimistic
or even meaningless. In this article, we derive a new, direct estimate of the
entrywise approximation error that is applicable in such cases. Our argument is
based on random embeddings, the tensor-structured Hanson--Wright inequality,
and the core coherences of a tensor. The theoretical results are accompanied
with numerical experiments: we compute low-rank tensor-train approximations in
the maximum norm for two classes of tensors using the method of alternating
projections.
