TITLE: Continuous extension of the limit functional
QUESTION [4 upvotes]: Let $(\ell^{\infty})'$ be the $\mathbb{F}$-vector space of linear and continuous (bounded) functionals $\ell^{\infty}\rightarrow \mathbb{F}$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$ (but we can assume $\mathbb{F}=\mathbb{R}$, if needed) and $\ell^{\infty}$ has the sup norm $\parallel\cdot\parallel_{\infty}$. Let also $c$ be the subspace of $\ell^{\infty}$ consisting of convergent sequences. Then the limit functional $\lim\colon c\rightarrow \mathbb{F}$ sending a convergent sequence to its limit is a continuous, linear functional with operator norm $1$ ($c$ has the sup norm as well).
I am asked to prove or disprove that there exist distinct elements $f,g\in(\ell^{\infty})'$ which extend the limit functional on $c$. I think the claim is true, but, up to now, I have been able to prove only the following fact (for $\mathbb{F}=\mathbb{R}$), using Hahn-Banach's extension Theorem: for every real number $\lambda$ with $-1\leq \lambda \leq 1$, there exists a linear extension $h_{\lambda}$ of the limit functional to the whole $\ell^{\infty}$ such that, for all $\alpha\in\mathbb{R}$ and any convergent sequence $x\in c$, if $y$ is the sequence $((-1)^{n})_{n\in\mathbb{N}}$, then $$h_{\lambda}(ay+x)=a\lambda +\lim(x)\leq \limsup(ay+x).$$
In particular, there are uncountably many linear extension of the limit functional. I can not prove that at least two of these are continuous though.
Can someone help me solving this problem with a worked solution? (I have looked for Banach limits around, but I have not found an explicit proof of non uniqueness of such continuous extensions of the limit extension).
Thanks in advance.

REPLY [1 votes]: The key observation is that, in the OP's notation,
$$\tag{1}
\|ay+x\|_\infty\geq\max\{|a|,|\lim x|\}.
$$
(proof below). Then
$$
|h_\lambda(ay+x)|=|\lambda a+\lim x|\leq|\lambda|\,|a|+|\lim x|\leq(|\lambda|+1)\|\lambda a y + x\|_\infty.
$$
Thus $h_\lambda$ is bounded and now you can extend by Hahn-Banach. By construction $h_\lambda\ne h_\lambda'$ if $\lambda\ne\lambda'$. 
Proof of (1). This follows from the fact that, for $n$ big enough, $ay_n+ x_n$ is alternatively very near $a+\lim x$ and $-a+\lim x$ (because $x_n$ is very near $\lim x$). More explicitly:
Claim.
$$
\max\{|\lambda+\mu|,|\lambda-\mu|\}\geq\sqrt{|\lambda|^2+|\mu|^2}\geq\max\{|\lambda|,|\mu|\}.
$$
To prove the claim, assume first that $\lambda\in\mathbb R+$, $\mu=re^{it}$. Then
$$
|\lambda\pm re^{it}|^2=(\lambda+r\cos t)^2+r^2\sin^2t=\lambda^2+r^2\pm 2\lambda r\cos t.
$$
Since either $\cos t\geq0$ or $\cos t<0$, one of the two choices of sign puts the expression above $\lambda^2+r^2=|\lambda|^2+|\mu|^2$. 
When $\lambda=se^{iv}$, $|\lambda+r^{it}|=|s+re^{i(t-v)}|$ and we can proceed as above.
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