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)$\mathcal {F}(\Omega)$ of scalar‐valued functions on a nonempty set Ω$\Omega$ is said to admit a strong linearization if there are a locally convex Hausdorff space Y$Y$, a map δ:Ω→Y$\delta: \Omega \rightarrow Y$, and a topological isomorphism T:F(Ω)→Yb′$T: \mathcal {F}(\Omega)\rightarrow Y_{b}^{\prime }$ such that T(f)∘δ=f$T(f)\circ \delta = f$ for all f∈F(Ω)$f\in \mathcal {F}(\Omega)$. 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)$\mathcal {F}(\Omega)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces.