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consider the space of Natural numbers equipped with the counting measure.

find a bounded linear functional $\Phi \in (l^\infty)^*$ which is not of the form $\Phi=\Phi_g$ for some $g \in l^1$

Not sure where to start looking, any hints?

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i dont think you can explicitly find any, you need some kind of existence theorem mathoverflow.net/questions/5351/… –  yoyo Apr 5 '11 at 22:07
    
@yoyo: That depends what you mean by "explicit". A limit along a nonprincipal ultrafilter isn't the most explicit thing in the world, but I think it's more explicit than just using Hahn–Banach. –  Chris Eagle Apr 5 '11 at 22:30
    
give one example of a non principal ultrafilter... –  yoyo Apr 6 '11 at 2:12
    
@Chris, are you talking about something like en.wikipedia.org/wiki/Ultralimit#Ultrafilters? –  user9024 Apr 6 '11 at 17:46
    
@Jenny: the section "Limit of a sequence of points with respect to an ultrafilter" is what I'm talking about. –  Chris Eagle Apr 6 '11 at 17:49

2 Answers 2

up vote 5 down vote accepted

You will need the Hahn-Banach theorem. In the following $x=(x_k)_{k=1}^\infty$ represents a sequence.

  1. Consider the subspace $c$ of all convergent sequences. Define $\Phi_0\colon c\to \mathbb{R}$ (or $\mathbb{C}$) by $\Phi_0(x)=\lim_{k\to\infty}x_k$.
  2. Show that $\Phi_0$ is bounded on $c$ with norm equal to $1$.
  3. Use the Hahn-Banach theorem to extend it to a bounded linear functional on $\ell^\infty$.
  4. Show that there is no sequence $a\in\ell^1$ such that $\Phi(x)=\sum_{k=1}^\infty a_kx_k$ for all $x\in\ell^\infty$. (Consider $\Phi$ on $c_0$, the space of sequences converging to zero.)
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This opens the gate to Banach generalized limits. I believe that none of these can be represented by an $\ell^1$ element. –  ncmathsadist Apr 22 '11 at 3:34

To expand on Chris Eagle's suggestion (since it seems to be somewhat confusing to the OP):

An ultrafilter $\mathcal{F}$ on $\mathbb{N}$ is a collection of subsets of $\mathbb{N}$ such that:

  1. $\emptyset\notin \mathcal{F}$;
  2. If $A\in\mathcal{F}$ and $A\subseteq B\subseteq\mathbb{N}$, then $B\in\mathcal{F}$;
  3. If $A,B\in\mathcal{F}$, then $A\cap B\in\mathcal{F}$;
  4. For every $X\subseteq\mathbb{N}$, either $X\in\mathcal{F}$ or $\mathbb{N}-X\in\mathcal{F}$.

For example, $\mathcal{F}=\{A\subseteq \mathbb{N}\mid 0\in A\}$ is an example of an ultrafilter; this is a principal ultrafilter: an ultrafilter that consists of "all subsets that contain" a given element.

Using Zorn's Lemma one can show that there exist ultrafilters on $\mathbb{N}$ that are not principal. These are often useful for defining structure, especially on products indexed by $\mathbb{N}$.

Consider $\ell^0$, the set of all bounded real sequences. Given a sequence $(x_n)$, a point $x$, and an $\epsilon\gt 0$, we can consider the $\epsilon$ ball around $x$, $B(x,(x_n),\epsilon) = \{n\in\mathbb{N}\mid |x_n-x|\lt \epsilon\}$.

We say that the sequence $(x_n)$ $\mathcal{F}$-converges to $x$ if and only if for every $\epsilon\gt 0$, $B(x,(x_n),\epsilon)\in\mathcal{F}$.

Then:

  • If $\mathcal{F}$-limit, if it exists, is unique:

    Suppose that $x\neq y$ and $B(x,(x_n),\epsilon)\in\mathcal{F}$ for all $\epsilon\gt 0$. Let $\delta=\frac{1}{2}|x-y|$. If $|x_n-y|\lt\delta$, then $|x_n-x|\geq \delta$, since $2\delta=|x-y|\leq |x-x_n|+|x_n-y| = |x-x_n|+\delta$. Therefore, $B(y,(x_n),\delta) \subseteq \mathbb{N}-B(x,(x_n),\delta))$. In particular, we cannot have $B(y,(x_n),\delta)\in\mathcal{F}$: if $B(y,(x_n),\delta)\in\mathcal{F}$, then its intersection with $B(x,(x_n),\delta)$ would also be in $\mathcal{F}$, which is impossible because the intersection is empty. So $(x_n)$ does not converge to $y$.

  • If $\lim_{n\to\infty}x_n = x$ in the usual sense, then $(x_n)$ will $\mathcal{F}$-converge to $x$.

    Let $\epsilon\gt 0$. Then there exists $N$ such that for all $n\geq N$, we have $|x_n-x|\lt \epsilon$. In particular, $\mathbb{N}-B(x,(x_n),\epsilon)\subseteq \{1,2,\ldots,N-1\}$. But $X=\{1,2,\ldots,N-1\}\notin \mathcal{F}$, because no singleton is in $\mathcal{F}$ (since $\mathcal{F}$ is nonprincipal), so no finite subset of $\mathcal{N}$ is in $\mathcal{F}$ (the complement of every singleton is in $\mathcal{F}$ since $\mathcal{F}$ is an ultrafilter, so any finite intersection of complements of singletons is in $\mathcal{F}$; this means that the complement of any finite subset of $\mathbb{N}$ is in $\mathcal{F}$, so no final subset is in $\mathcal{F}$).

    Since the complement of $B(x,(x_n),\epsilon)$ is finite, it follows that $B(x,(x_n),\epsilon)\in\mathcal{F}$. This holds for all $\epsilon$, so $(x_n)$ also $\mathcal{F}$-converges to $x$.

  • If $(x_n)$ $\mathcal{F}$-converges to $x$, then for any real number $\alpha$, $(\alpha x_n)$ $\mathcal{F}$-converges to $\alpha x$.

    This is clear if $\alpha=0$, since then $(\alpha x_n)$ is the constant sequence $0$ which converges to $0$; assume $\alpha\neq 0$. Let $\epsilon\gt 0$. Then $$\begin{align*} B(\alpha x,(\alpha x_n),\epsilon) &= \{n\in\mathbb{N}\mid |\alpha x -\alpha x_n|\lt\epsilon\}\\ &= \{n\in\mathbb{N}\mid |x-x_n|\lt\epsilon/|\alpha|\}\\ &= B(x,(x_n),\epsilon/|\alpha|)\in\mathcal{F}, \end{align*}$$ so $(\alpha x_n)$ will $\mathcal{F}$-converge to $\alpha x$.

  • If $(x_n)$ $\mathcal{F}$-converges to $x$ and $(y_n)$ $\mathcal{F}$-converges to $y$, then $(x_n+y_n)$ will $\mathcal{F}$-converge to $x+y$.

    Let $\epsilon\gt 0$. Then $B(x,(x_n),\epsilon/2), B(y,(y_n),\epsilon/2)\in\mathcal{F}$, hence their intersection lies in $\mathcal{F}$. If $|x_n-x|\lt\epsilon/2$ and $|y_n-y|\lt\epsilon/2$, then $|(x_n+y_n)-(x+y)|\lt\epsilon$; so $$B(x+y, (x_n+y_n),\epsilon) \supseteq B(x,(x_n),\epsilon/2)\cap B(y,(y_n),\epsilon/2) \in \mathcal{F}$$ so $B(x+y, (x_n+y_n),\epsilon)\in\mathcal{F}$. Therefore, $(x_n+y_n)$ $\mathcal{F}$-converges to $x+y$.

  • The big nontrivial part which I won't prove: if $(x_n)$ is bounded, then $x_n$ has an $\mathcal{F}$-limit. I won't be able to prove it here, but it's nonetheless true. (It follows from the fact that if your sequences take values in a compact metric space, then they necessarily have an $\mathcal{F}$-limit).

So, taking these facts, define $\Phi_{\mathcal{F}}\colon\ell^{\infty}\to\mathbb{R}$ by letting $\Phi_{\mathcal{F}}(x_n)$ be the (unique) $\mathcal{F}$-limit of $(x_n)$. This is a linear functional. Now show that it is not equal to $\Phi_g$ for any $g\in\ell^1$.

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