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
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.
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