In the proof of proposition 3.23 (bellow in the picture) in Engel-Nagel book's (One-Parameter Semigroups for Linear Evolution Equations) they claim that $z'_\lambda$ (in the sphere of $X^\ast$) has a accumulation point as $\lambda \to 0$ in weak* topology (last line of figure). My doubt is if we need assume that our space is separable to garante that the closed dual unit ball $B_X*$ is metrizable with the weak* topology and then get the limit point of a sequence $z'_{\lambda_n}$. In the book this is not assumed separability of $X$. There is any other result which could be used for which the separability hypothesis is not necessary?
