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 ∘ U−1. 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 ∘ T−1= r1∗ for some norm r1 on Y
equivalent to r, but such that q ∘ U−1≠s1∗ for any norm s1 on Z equivalent to
s.
