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.

2 thoughts on “A problem with sequences”

Is this because $(l^\infty)^*$ is not weak- $*$ metrizable? So Compactness does not imply sequential compactness

gshakan says:

March 28, 2016 at 8:02 pm

Nice Amir, that’s precisely the issue.

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A problem with sequences

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2 thoughts on “A problem with sequences”

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