whether this set is closed subset of a Hilbert space Whether the set of sequences $(x_n \in l_2: \sum_{n=1}^{\infty}\frac{x_n}{n}=1 )$ is closed ? How do you find the limit point of a set of sequences?  Moreover what is the complement of this set?, and whether that is open? (IIT-GATE 2015)

We have $f=(1,{1 \over 2},..., {1 \over n},...) \in l_2$, hence
$f^*(x)= \langle f,x\rangle$ is a continuous linear functional, hence the
inverse image of a closed set is closed.
The complement is just the set $\{x | f^*(x) \neq 1 \}$ which is open
because its complement is closed (or, indeed, because $\mathbb{R} \setminus \{1\}$ is open).
Another way is to note that $H=\{x | f^*(x) = 1 \} = \{e_1\}+\ker f^*$, that is,
$H$ is just a translate of the kernel, which is closed.
