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A characterization of real and complex Hilbert spaces among all normed spaces
Published online by Cambridge University Press: 17 April 2009
Abstract
Let X be a real or complex normed space and L(X) the algebra of all bounded linear operators on X. Suppose there exists a *-algebra B(X) ⊂ L(X) which contains the identity operator I and all bounded linear operators with finite-dimensional range. The main result is: if each operator U ∈ B(X) with the property U*U = UU* = I has norm one then X is a Hilbert space.
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- Copyright © Australian Mathematical Society 1983