  Crouzeix and Palencia recently showed that the numerical range of a
Hilbert-space operator is a $(1+\sqrt2)$-spectral set for the operator. One of
the principal ingredients of their proof can be formulated as an abstract
functional-analysis lemma. We give a new short proof of the lemma and show
that, in the context of this lemma, the constant $(1+\sqrt2)$ is sharp.
