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