What is an example of infinite dimensional subspace that is not closed? In a theorem I am reading about closed subspace the author states that an infinite dimensional subspace need not be closed.
What is an example of infinite dimensional subspace that is not closed?

Let $\ell^2$ be the space of all square-summable real (or complex) sequences $x = (x_1,x_2, \ldots)$ with norm $\|x\| = \displaystyle ( \sum |x_i|^2)^{1/2}$. Let $V \subset \ell ^2$ be the subspace of all sequences with all but finitely many entries equal to zero. Then $V$ is infinite-dimensional but not closed. It is not closed because its closure contains the limit point $(1,1/2, 1/3, \ldots)$
