Dual of $l_\infty$

Question

Let $l_\infty$ be supplied with the supremum norm and $(l_\infty)^*$ be its dual. Further, let $e_n$ be the orthonormal basis of $e_n=(0,0,...,1,...,0)$.

Are the following statements true or false?
1. there exists a $\phi\in(l_\infty)^*$ with $\phi\neq0$ and $\phi(e_n)=0$ for all $n\in\mathbb{N}_{>0}$
2. there exists a $\phi\in(l_\infty)^*$ with $\phi(a)=\lim_{n\rightarrow\infty}a_n$ for all $a=(a_n)\in c$
3. there exists a $\phi\in(l_\infty)^*$ with $\phi(a)=\sum_{n=1}^{\infty}a_n$ for all $a=(a_n)\in l_1$

I do not know how to begin this proof and would really appreciate any hint to get me starting. Thank you.

Answer 1

For 2. consider $\psi\colon c\to \mathbb{F}$ given by $\psi(a)=\lim_{n\to\infty}a_n$. Prove that it is bounded, and use the Hahn-Banach theorem to extend it to $\ell^\infty$.

The example constructed in 2. shows that the answer to 1. is yes.

For 3. let $x_n=\sum_{k=1}^ne_k$. Then, if there were such a $\phi$, we would have $$ \phi(x_n)=n\le C\,\|x_n\|_\infty=C $$ for some constant $C$.

Answer 2

Think about what conditions $ \phi \in l_\infty^*$ has to fullfill. How is the dual defined? Then check if this is possible when $\phi$ fullfills 1), 2) or 3).