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I'm reading the proof of Banach Alaoglu using nets. The theorem states that the closed unit ball in $X^\ast$ is weak star compact.

My question is: If $\varphi_\alpha$ is a universal net mapping into the closed unit ball why is $\varphi_\alpha (x)$ also universal?

A set is called universal iff for any set $Y$ it is eventually in $Y$ or in $Y^c$. The net $\varphi_\alpha(x)$ is a map $A \to \mathbb C$ (if $X$ is acomplex normed vector space).

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The image of a universal net under any map $f\colon E \to F$ is always universal.

That follows, since $f^{-1}(F\setminus C) = E \setminus f^{-1}(C)$, so $f(\varphi_\alpha)$ will be eventually in $C$ if $\varphi_\alpha$ is eventually in $f^{-1}(C)$, and it will be eventually in $F\setminus C$ otherwise, since then $\varphi_\alpha$ is eventually in $E\setminus f^{-1}(C) = f^{-1}(F\setminus C)$.

It is analogous for filters, the image of an ultrafilter is always an ultrafilter.

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Take $$Z_Y := \{\psi \in X^\ast \mid \psi(x) \in Y\}.$$

Note $(Z_Y)^c = Z_{Y^c}$ and apply the universality of $\phi_\alpha$.

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