# How do I show that the following is a basis for the weak topology on $X$?

$S(x_0, \epsilon, A^*)=\{x \in X : |f(x) - f(x_0)|< \epsilon \, \forall \, f \in A^* \}$ Where $A^*$ is a finite subset of the dual space $X'$.

I'm stuck on how to apply the two conditions of basis on the set.

• Isn't this the definition of the weak topology? What is your definition? – T. Eskin Jun 7 '15 at 15:17
• @ThomasE. The definition I'm using is that the topology on $X$ defined by the semi norm $\{pf : f \in X^*\}$ is the weak topology. – Mark Jun 7 '15 at 15:52
• What is $p$ and what do you mean by that set being a semi-norm? – T. Eskin Jun 7 '15 at 15:56
• @Mark: In your previous comment did you mean $\{p_f:f\in X^*\}$, where $p_f(x)=|f(x)|$? – Brian M. Scott Jun 7 '15 at 16:03
• Maybe you'll find this thread useful: math.stackexchange.com/questions/305808/… – T. Eskin Jun 7 '15 at 16:43