by H. Cheng, Z. Gimbutas, P.G. Martinsson, V. Rokhlin:

- Before Theorem 1 there is a sentence: "The theorem also states
that the first k columns of AP form a well-conditioned basis for the
column space of A to within accuracy $\varepsilon$." This sentence
might be misleading; what the theorem really says is that any vector
in the column space of A can be expressed as a linear combination of
the first k columns of AP in such a way that no expansion coefficient
is large. However, it is not necessarily the case the that the first k
columns of AP form a well-conditioned matrix. (Indeed, this is
typically
**not**the case in the common situation that A is a discrete representation of a compact operator.) - The comments in 1 also apply to the last sentence of Remark 2.