Metrization theory and the Kadec property

Abstract

The uniform structure of a descriptive normed space $(X,\Vert \cdot \Vert )$ always admits a description with an $\left(F\right)$-norm $\Vert \cdot {\Vert }_1$ such that weak and norm topologies coincide on

$$\{x\in X:\Vert x{\Vert }_1=\rho \}$$ for every $\rho >0$.