In $l^2=\{ (a_n): \sum_{n=1}^{\infty}\vert a_n^2\vert< \infty\}$ which of the following are true?

Question

In $l^2=\{ (a_n): \sum_{n=1}^{\infty}\vert a_n^2\vert< \infty\}$ which of the following are true?

1) Every bounded sequence in $l^2$ has a convergent subsequence.

2) $l^2$ has a proper closed subspace.

3) There exist a non zero continuous linear functional on $l^2$

4) If $(x_n)$ is a Cauchy sequence in $l^2$, then the sequence $(f(x_n))$ is Cauchy for every bounded linear functionals $f$ on $l^2$

My work:

4) is true if I take $\epsilon_1=\epsilon /\vert\vert f \vert \vert$

but for other option I find difficulties, Please help.

Answer 1

1. False. Otherwise, the unit ball $\ell^2$ would be compact, but it isn't.
2. True. Take, for instance, those sequences in $\ell^2$ such that $a_n=0$ when $n>1$.
3. True. For each $n\in\mathbb N$, lete $e(n)\in\ell^2$ be such that $e(n)_n=1$ and that $e(n)_k=0$ if $k\neq n$. Extend $\{e(n)\,|\,n\in\mathbb{N}\}$ to a basis $B$ of $\ell^2$ and consider the linear functional $\alpha$ in $\ell^2$ such that $(\forall n\in\mathbb{N}):\alpha\bigl(e(n)\bigr)=n$ and that $\alpha$ takes the value $0$ on every other element of $B$.
4. True, and your approach is good, but not particulary well explained.