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} = r_{1}^{∗} for some norm r_{1} on Y
equivalent to r, but such that q ∘ U^{−1}≠s_{1}^{∗} for any norm s_{1} on Z equivalent to
s.
