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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 1, Pages 47-53

An algebraic characterization of complete inner product spaces

Universitat Konstz, Fakultat fur Mathematik, Posfach 55 60, Konstanz 1 D-7750, Germany

Received 1 October 1983

Copyright © 1986 Hindawi Publishing Corporation. 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.


We present a characterization of complete inner product spaces using en involution on the set of all bounded linear operators on a Banach space. As a metric conditions we impose a “multiplicative” property of the norm for hermitain operators. In the second part we present a simpler proof (we believe) of the Kakutani and Mackney theorem on the characterizations of complete inner product spaces. Our proof was suggested by an ingenious proof of a similar result obtained by N. Prijatelj.