How to conclude that $\ell_\infty$ is not separable from this exercise?

by Filburt   Last Updated January 16, 2018 13:20 PM

I have done an exercise that goes like this:

Consider the operator $\Phi: \ell_1\to\ell_\infty$ that associates each $x=(x_j)_j\in\ell_1$ to $\Phi (x)\in (\ell_\infty)'$ given by $\Phi(x)(y)=\sum x_j y_j$, for all $y=(y_j)_j\in\ell_\infty$. Show that $\Phi$ is well defined, is linear and bounded. Construct an element $g\in (\ell_\infty)'\backslash \ \Phi(\ell_1)$. Is $\ell_\infty$ separable?

So... I went through all this exercise, but not the final. How can I conclude that $\ell_\infty$ is not separable from this? Or is this question not related to the exercise (what doesn't make sense)?

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