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We study the problem of existence and uniqueness of isometric Banach predualsof a Banach space. We derive necessary and sufficient conditions for theexistence of an isometric Banach predual of a Banach space $X$. Then we focuson the case that $X=\mathcal{F}(\Omega)$ is a Banach space of scalar-valuedfunctions on a non-empty set $\Omega$ and describe those spaces which admit aspecial isometric Banach predual, namely a \emph{strong isometric Banachlinearisation}, i.e. there is a Banach space $Y$, a map $\delta\colon\Omega\toY$ and an isometric isomorphism $T\colon\mathcal{F}(\Omega)\to Y^{\ast}$ suchthat $T(f)\circ \delta= f$ for all $f\in\mathcal{F}(\Omega)$. Finally, we givenecessary and sufficient conditions for Banach spaces $\mathcal{F}(\Omega)$with a strong isometric Banach linearisation to have a (strongly) uniqueisometric Banach predual.

On linearisation, existence and uniqueness of preduals: The isometric case