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There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in particular, of the Bessaga-Pelczynski selection principle which, given a Schauder basis $(e_n)$ for a Banach space $X$, allows the passage from a normalized sequence $(x_n)$ in $X$ which converges weakly to $0$, to a subsequence $(x_{n_k})$ which is congruent to a block basic sequence.

What I am wondering is if, given a non-principle ultrafilter $\mathcal{U}$ on $\mathbb{N}$, and a sequence $(x_n)$ of unit vectors weakly converging to 0, can we find such a subsequence $(x_{n_k})$ as above so that the index set $\{n_k:k\in\mathbb{N}\}$ is in $\mathcal{U}$?

I would also be interested any related selection principles which can be done "along an ultrafilter", or reasons why one cannot hope for results like this.

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in particular, of the Bessaga-Pelczynski selection principle which, given a Schauder basis $(e_n)$ for a Banach space $X$, allows the passage from a normalized sequence $(x_n)$ in $X$ which converges weakly to $0$, to a subsequence $(x_{n_k})$ which is congruent to a block basic sequence.

What I am wondering is if, given a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, and a sequence $(x_n)$ of unit vectors weakly converging to 0, can we find such a subsequence $(x_{n_k})$ as above so that the index set $\{n_k:k\in\mathbb{N}\}$ is in $\mathcal{U}$?

I would also be interested any related selection principles which can be done "along an ultrafilter", or reasons why one cannot hope for results like this.

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in particular, of the Bessaga-Pelczynski selection principle which, given a Schauder basis $(e_n)$ for a Banach space $X$, allows the passage from a normalized sequence $(x_n)$ in $X$ which converges weakly to $0$, to a subsequence $(x_{n_k})$ which is congruent to a block basic sequence.

What I am wondering is if, given a non-principle ultrafilter $\mathcal{U}$ on $\mathbb{N}$, can we find such a subsequence $(x_{n_k})$ as above so that the index set $\{n_k:k\in\mathbb{N}\}$ is in $\mathcal{U}$?

I would also be interested any related selection principles which can be done "along an ultrafilter", or reasons why one cannot hope for results like this.

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in particular, of the Bessaga-Pelczynski selection principle which, given a Schauder basis $(e_n)$ for a Banach space $X$, allows the passage from a normalized sequence $(x_n)$ in $X$ which converges weakly to $0$, to a subsequence $(x_{n_k})$ which is congruent to a block basic sequence.

What I am wondering is if, given a non-principle ultrafilter $\mathcal{U}$ on $\mathbb{N}$, and a sequence $(x_n)$ of unit vectors weakly converging to 0, can we find such a subsequence $(x_{n_k})$ as above so that the index set $\{n_k:k\in\mathbb{N}\}$ is in $\mathcal{U}$?

I would also be interested any related selection principles which can be done "along an ultrafilter", or reasons why one cannot hope for results like this.