International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 611-614
Research notes

A note on best approximation and invertibility of operators on uniformly convex Banach spaces

James R. Holub

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061, Virginia, USA

Received 1 October 1990

Copyright © 1991 James R. Holub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It is shown that if X is a uniformly convex Banach space and S a bounded linear operator on X for which IS=1, then S is invertible if and only if I12S<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.