On linearization and uniqueness of preduals

Abstract

We study strong linearizations and the uniqueness of preduals of locally convex Hausdorff spaces of scalar-valued functions. Strong linearizations are special preduals. A locally convex Hausdorff space $F\left(\Omega \right)$ of scalar-valued functions on a nonempty set $\Omega$ is said to admit a strong linearization if there are a locally convex Hausdorff space $Y$, a map $\delta :\Omega \to Y$, and a topological isomorphism $T:F\left(\Omega \right)\to Y_b^{\prime }$ such that $T\left(f\right)\circ \delta =f$ for all $f\in F\left(\Omega \right)$. We give sufficient conditions that allow us to lift strong linearizations from the scalar-valued to the vector-valued case, covering many previous results on linearizations, and use them to characterize the bornological spaces $F\left(\Omega \right)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.