Preamble: let $(X, \tau)$ be a topological vector space over $\mathbb K$, and $X'$ its (topological) dual space. Then, as I understand, the family of sets given by $$ \mathcal B = \{ f^{-1} (V) \mid f \in X' , \, V \text{ open in } \mathbb K \} $$ is a subbasis for the weak topology induced by $X'$ on $X$. This topology, of course, has the property of being the coarsest topology on $X$ for which every $f \in X'$ is $\tau$-continuous.
Question: Is it possible to make an analogous characterization of the weak-* topology induced by $X$ on $X'$? In particular, is it true that the family of sets given by $$ \mathcal E = \{ \hat x ^{-1} (V) \mid \hat x \in X'' , \, V \text{ open in } \mathbb K \} $$ is a subbasis for the weak-* topology?
Any input will be much appreciated, I am quite lost here...