Sequences and nets

According to the (Banach-)Alaoglu theorem, for any Banach space X the closed unit ball of X^* is compact in the weak* topology (the weakest/coarsest topology that makes all evaluation functionals f\mapsto f(x) continuous).

For example, take X=\ell_{\infty}. The dual space \ell_{\infty}^* contains an isometric copy of \ell_1 because \ell_\infty^*=\ell_1^{**}. The sequence x_n=n^{-1}(e_1+\dots+e_n), where e_n are the standard basis vectors, is contained in the unit sphere of \ell_1. Should (x_n) have a weak*-convergent subsequence? Maybe it should, but it does not.

Indeed, take any subsequence (x_{n_k}). If necessary, choose a further subsequence so that n_{k+1}\ge 3n_k for all k. Define

\displaystyle y=\sum_{k}(-1)^k\sum_{n_{k-1}< j\le n_k} e_j

where we set n_0=0. Two things to notice here: (1) y\in \ell_{\infty}; and (2) at least 2/3 of the coefficients of e_j, 1\le j\le n_k, have the sign (-1)^k. Hence, \langle x_{n_k},y\rangle\le -1/3 when k is odd and \langle x_{n_k},y\rangle\ge 1/3 when k is even. This shows that (x_{n_k}) does not converge in the weak* topology.

The above does not contradict the Banach-Alaoglu theorem. Since \ell_\infty is not separable, the weak* topology on the unit ball of its dual is not metrizable. The compactness can be stated in terms of nets instead of sequences: every bounded net in X^* has a convergent subnet. In particular, the sequence (x_n) has a convergent subnet (which is not a sequence). I personally find subnets a recipe for erroneous arguments. So I prefer to say: the infinite set \{x_n\} has a cluster point x; namely, every neighborhood of x contains some x_n. You can use the reverse inclusion of neighborhoods to define a subnet, but I’d rather not to. Everything we want to know about x can be easily proved from the cluster point definition. For example,

  • \|x\|_{X^*}=1
  • \langle x, 1_{\infty}\rangle =1 where 1_{\infty} stands for the \ell_\infty vector with all coordinates 1.
  • \langle x, y\rangle = \langle x, Sy\rangle where S is the shift operator on \ell_\infty

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