Reading up on Banach limits I was thinking if this property is kept for the extension of a limit. I couldn't find any proof so I wanted to see if such an example exists at least.
On the space $l_\infty$ is there a Banach limit such that for any bijection $\phi: \mathbb{N} \to \mathbb{N}$, and any sequence $a_n$ in $l_\infty$ we have that $Lim(a_n)=Lim(a_{\phi(n)})$, where $Lim$ is a Banach limit $Lim:l_\infty \to \mathbb{R}$.
I tried to construct such a Banach limit, taking the sublinear functional $p(x_n)=\limsup_{n\to\infty}x_n$ and $l(x_n) = lim_{n->\infty} x_n$.
Knowing that $lim(x) \leq p(x)$ (equal actually) for convergent sequences, and it is a linear functional, so using Hahn-Banach it can be extended to a linear functional $Lim(x_n)$ such that $Lim(x_n)\leq p(x_n)$ for any $x_n \in l_\infty$.
Now I know that this functional is shift-invariant and that $\liminf_{n\to\infty}x_n\leq L(x_n)\leq limsup_{n\to{\infty}}x_n$.
Let $\phi: \mathbb{N} \to \mathbb{N}$ be a bijection, so we have $ Lim(a_{\phi(n)})\leq\limsup_{n\to\infty}a_{\phi(n)}$
So $Lim(a_{\phi(n)})-Lim(a_n)\leq \limsup_{n\to\infty}a_{\phi(n)} - \liminf_{n\to\infty}a_n $ but this from this I can't find a way to show that they are equal. At most that the difference is between the difference of the upper and lower bounds of $a_n$.
Is this statement true, and how can it be proven, if not is there a linear functional that extends the limit with this property?
There are two cases to consider:
One where there’s a finite permutation, and one where there’s an infinite permutation.
Finite: the bijection, $\phi$ only affects the first $n$ terms of the sequence. Then $Lim(x)=Lim(S^n x)$, which deletes the finite permutation.
Infinite: A conditionally convergent series is one that is subject to the Riemann rearrangement theorem where an infinite permutation can change the limit to a different value.
Seeing as the normal limit itself doesn’t have the second property, not function can extend it and also have it.
The above answer only uses the axioms of the Banach limit and thus applies to all of them.