The standard HOSVD truncation of tensors yields a controlled error with respect to the
corresponding Hilbert norm, usually L2-norm. If the tensor represents a multivariate
function, the aim may be to evaluate the function at certain points. Therefore, we need
maximum error estimates instead. Usually, the maximum norm cannot be estimated
by the L2-norm. We show that, nevertheless, this is possible for the HOSVD truncated tensors.