It is shown that if X is a uniformly convex Banach space and S a bounded linear operator on
X for which ‖I−S‖=1, then S is invertible if and only if ‖I−12S‖<1. From this it follows that
if S is invertible on X then either (i) dist(I,[S])<1, or (ii) 0 is the unique best approximation to
I from [S], a natural (partial) converse to the well-known sufficient condition for invertibility that
dist(I,[S])<1.