A problem with sequences


In the above pdf, we address the following issue.

Consider the canonical embedding of j:  \ell^1(\mathbb{N})  \to \ell^{\infty}(\mathbb{N}) ^*. It is easy to see that no sub-sequence of j(e_1) , j(e_2) , \ldots can weak-star converge, yet Banach-Alaoglu asserts that the unit ball of \ell^{\infty}(\mathbb{N}) ^* is weak-star compact. So what is the problem?

This is the result of several discussions with Chris Gartland who explained to me a large portion of what is in the file.


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