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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...

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The weak-* topology is the coarsest topology such that the maps f_x:X'\to \mathbb{K} with f_x(x') = x'(x) are continuous (see, e.g. Wikipedia). Hence, your familiy $\mathcal{E}$ is too large (roughly speaking: you take all \hat x\in X'' but the $x\in X$ are enough).

What you have written is a subbasis of the weak topology on X' which may be finer than the weak-* topology on X' (see Vobo's comment below).

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    I think I get it now, thank you both!2011-04-07