The properties of $c_{00}$.

Question

I am new to functional analysis. I wondering what properties of $c_{00}$ have.

In particular, I am wondering:

1. Why is $c_{00}$ is not closed in $\ell_p$ for $p\in[1,\infty]$.

2. What does the dual space of $c_{00}$ look like?

3. Why the dual space of $c_{00}$ is reflexive?

Thank you in advance!

Answer 1

Question 1

The sequence $$ (u_n)=\begin{cases} 1/k^{2/p} & 1\le k \le n\\ 0 & k>n \end{cases}$$ is a sequence of elements of $c_{00}$ that is converging in $\ell_p$ to the sequence $(1/k^{2/p})$ that is not in $c_{00}$. Hence $c_{00}$ is not closed.

For the other questions, see the commentS of s.sharp above.