  We relate the problem of best low-rank approximation in the spectral norm for
a matrix $A$ to Kolmogorov $n$-widths and corresponding optimal spaces. We
characterize all the optimal spaces for the image of the Euclidean unit ball
under $A$ and we show that any orthonormal basis in an $n$-dimensional optimal
space generates a best rank-$n$ approximation to $A$. We also present a simple
and explicit construction to obtain a sequence of optimal $n$-dimensional
spaces once an initial optimal space is known. This results in a variety of
solutions to the best low-rank approximation problem and provides alternatives
to the truncated singular value decomposition. This variety can be exploited to
obtain best low-rank approximations with problem-oriented properties.
