On linearization and uniqueness of preduals

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⁡(Ω) of scalar-valued functions on a nonempty set Ω is said to admit a strong linearization if there are a locally convex Hausdorff space Y, a map δ:Ω→Y, and a topological isomorphism T:F⁡(Ω)→Yb′ such that T⁡(f)∘δ=f for all f∈F⁡(Ω). 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⁡(Ω) with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.