Vol. 32, No. 3, 1970

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Conjugate space representations of Banach spaces

Emile B. Roth

Vol. 32 (1970), No. 3, 793–797

Let a linear homeomorphism T from a Banach space X onto the conjugate space Y of a Banach space Y be called a conjugate space representation of X. If T : X Y and U : X Z are two conjugate space representations of X, say that T and U are essentially different if there is no linear homeomorphism P from Y onto Z satisfying P = T U1. It is proven here that if a nonreflexive Banach space has one conjugate space representation, it has uncountably many essentially different conjugate space representations. A Banach space X with norm p will be denoted by (X, p) when it is important to emphasize the norm. The dual of p is the norm p defined on the conjugate space (X,p) of (X,p) by p(f) = sup{|f(x)| : x X and p(x) = 1}. It is proven here that if T : (X,p) (Y,r) and U : (X,p) (Z,s) are two essentially different conjugate space representations of (X,p), then there exists a norm q on X equivalent to p such that q T1 = r1 for some norm r1 on Y equivalent to r, but such that q U1s1 for any norm s1 on Z equivalent to s.

Mathematical Subject Classification
Primary: 46.10
Received: 5 November 1968
Published: 1 March 1970
Emile B. Roth