Linked Questions

15
votes
7answers
2k views

Applications of ultrafilters

I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters? I'm ...
9
votes
1answer
3k views

Basic facts about ultrafilters and convergence of a sequence along an ultrafilter

Could you help, please. I need the information about the ultrafilters, namely, any ideas how one can see that they exist and a proof of the fact that for any ultrafilter every sequence on a compact ...
15
votes
3answers
833 views

Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
13
votes
1answer
2k views

Nonnegative linear functionals over $l^\infty$

My purpose is a clarification of the role of the axiom of choice in constructing limits for bounded sequences. Namely, we want a linear functional of norm 1 defined on the space of all bounded complex ...
7
votes
1answer
1k views

Dual of $\ell_{\infty}$ is not $\ell_1$

As the title indicates I'm trying to show that $\ell_{\infty}^{*}$ is not $\ell_1$. I've shown that for p, q conjugate and finite we do indeed have $\ell_{p}^{*} = \ell_q$, with the correspondence ...
5
votes
1answer
1k views

$\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD

Let the base field be the real numbers or the complex numbers (I don't think it will matter). Let $(\ell^{\infty})'$ be the continuous dual of the Banach space $\ell^{\infty}$. Let $\: f : \ell^1 ...
2
votes
3answers
118 views

Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective

I am working on the dual spaces of sequence spaces, and I want to show that the map $$ \Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i $$ is not surjective. I have already ...
6
votes
1answer
235 views

Does existence of some (nice) non-trivial functionals in $\ell_\infty^*\setminus\ell_1$ give a free ultrafilter on $\omega$?

If we there is a free ultrafilter $\mathscr U$ on $\omega$ then $$\newcommand{\Ulim}{\operatorname{{\mathscr U}-lim}}f \colon x = (x_n) \mapsto \Ulim x_n$$ defines a functional belonging to $\ell_\...
3
votes
0answers
457 views

continuous linear functional on $l^{\infty}$ space

Let $l_{\infty}$ be the space of all bounded complex-valued sequences equipped with the supremum norm. Consider the natural standard basis $\{e_n\}_{n \in \mathbb{N}}$ of $l_{\infty}$. For any ...
4
votes
1answer
154 views

Exercise in Hahn-Banach Theorem; Finding linear functional $-p(-x)\leq f(x)\leq p(x)$

(The following exercises are in Kreyszig's book 218 page; EXE 10) I want to solve the following exercise : If $X=l^\infty$, let $p(x)=\lim\sup x_i $, which is sublinear. Then find a linear functional $...
1
vote
1answer
72 views

Problems involving reflexivity and weak / weak-* convergence

Consider the following two questions where throughout $X$ is a Banach space, $X'$ denotes its dual and $(f_n)_{n=1}^\infty\subset X'$. We also denote the canonical map $J_X:X\to X''$ where for all $f\...