TITLE: Linear functional and convergent series in $\ell^\infty$
QUESTION [5 upvotes]: Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm and let $c,c_0$ be the subspaces of sequences that are convergent, resp. convergent to zero. Show that:

The linear functional $\ell_0\colon c\rightarrow \mathbb{C}$ defined for $x = (x_n) \in c$ by
$$ \ell_0(x) = \lim_{n\rightarrow \infty} x_n$$
extends to a continuous functional on $\ell^\infty$
if $L$ denotes the set of all continuous extensions of the functional $\ell_0$ from (1), then a sequence $x = (x_n) \in  \ell^\infty$ belongs to $c_0$ iff
$$\ell(x) = 0 \;\; \forall \ell \in L$$
Describe $c$ in a similar way
My try:

(1): This follows by Banach limits.
(2): $(\Rightarrow)$ follows by extension
$(\Leftarrow)$ Here Im a bit unsure, assume $x \not \in c_0$ if $x \in c$ we get an contradiction. But if $x\not \in c$ what happens then, can we use a subsequence? since we have bounded functionals? can we use $\ell x = \lim_{k \rightarrow \infty} x_{n_k}$ or something like that, would $\ell \in L$?
(3): same as two I suppose, can we use subseqeunces?
Please correct what I'm missed

REPLY [2 votes]: In the general case you have the equivalence: $x\in c$ if and only if there exist a constant $a\in\mathbb{R}$ such that $l(x)=a$ for all $l\in L$.
One is easy, so suppose that $x\in \ell^\infty\setminus c$. Consider the space $$c+x\mathbb{R}=\{z\in\ell^\infty:\ z=y+\lambda x,\ y\in c,\ \lambda\in\mathbb{R} \}$$
Let $b\neq a$ and define a bounded linear functional $\tilde{l}:c+x\mathbb{R}\rightarrow\mathbb{R}$ by $$\tilde{l}(z)=l_0(y)+\lambda b$$
Note that $\tilde{l}$ is a bounded linear functional defined in a subspace of $\ell^\infty$ and such that $\tilde{l}(x)=b\neq a$. From here you can conclude.