TITLE: Dense subspace of $\ell_2$
QUESTION [7 upvotes]: How can I show that the subspace $\{x\in \ell_2 \mid \sum_{n=1} ^\infty x_n \frac 1{\sqrt n}=0\}$ is dense in $\ell_2$?
Can it be done with "simple" tools just by using the definition of a dense subspace? or can it be done by using the fact that $\ell_2$ is a Hilbert space?

REPLY [3 votes]: Let us note that your set $V$ is actually a vector subspace. It is enough to show that $V$ separates points, i.e., if for all $f\in V$ and some $x\in \ell_2$ we have $\langle f,x\rangle = 0$ then $x=0$.
We argue by contraposition. Take $x\in \ell_2$ and suppose that $x\neq 0$. Then $x_k\neq 0$ for some $k$. Let $f$ be such that $f_k = 1$, for some $m\neq k$, $$f_{m}=-\frac{1}{\sqrt k}\cdot\sqrt{m+1}$$
such that $-\sqrt{m+1} \cdot x_m \neq 1$ and $f_n=0$ otherwise. This is possible as the sequence $(1/\sqrt{k+1})_{k=1}^\infty$ is not in $\ell_2$. Then $f\in V$ and $\langle f,x\rangle\neq 0$.