An algebraic characterization of complete inner product spaces

Vasile I. Istratescu

doi:10.1155/s0161171286000066

Abstract

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.

Highlights

Let X be a complex Banach space and L (X)be the set of all bounded linear operators on X.There are conditions about [(X) (or parts of i(X)) which force X to be an inner product space,i.e, these conditions imply the exist-,ence of an inner product on X such that for all x in where li, IIll -< >I . denotes the norm of the Banach spaceOne of these sets of conditions was formulated by S.Kakutani and G.W.Mackey (1944,1946) and reads as follows:suDpose that on i(X) we have a mapping *whose values are in i(X) such that the following relations hold:(T+S)* =T*+ S* 2
We present a characterization of complete inn=r product spaces using en involution on the set of all bounded linear operators on a Banach space
In the second part we present a simpler proof of the Kakutani and Mackney theorem on the characterizations of complete inner product spaces

Summary

INTRODUCTION

Be the set of all bounded linear operators on X.There are conditions about [(X) (or parts of i(X)) which force X to be an inner product space,i.e, these conditions imply the exist-. One of these sets of conditions was formulated by S.Kakutani and G.W.Mackey (1944,1946) and reads as follows:suDpose that on i(X) we have a mapping *. P =PI +P2 is a hermitian projection on X.From our condition we get that the norm of P is 1. the Kakutani-Bohnenblust theorem implies that on the space X there exists an inner product satisfying the theorem.

Then clear we have the inclusion

Of fundamental importance of the above inner product is the property

Now if