TITLE: Example for a proper dense subspace?
QUESTION [6 upvotes]: I have been reading some books on functional analysis, and many of them keep talking about a vector space along with a dense proper subspace of it (especially when constructing counterexamples). But to me, it is kind of hard to imagine what such a dense proper subspace would look like, Nor am I convinced that such a structure actually exists at all. 
So can anybody help giving an example of it, or alternatively, give a proof that such a subspace actually exists (though I guess the existence would quite likely just follow from a construction)?
Thank you!!

REPLY [2 votes]: The space $Y$ of finitely supported sequences is a proper subspace of $\ell^p$ for $1 \leq p \leq \infty$. For $ 1 \leq p < \infty$, it is dense in $\ell^p$ with respect to the $\ell^p$-norm. To show this, consider an arbitrary $x = (\xi_1, \xi_2, \ldots) \in \ell^p$ and consider the sequence $x_n = (\xi_1, \xi_2, \ldots, \xi_n, 0, 0, \ldots) \in Y$. It's easy to see that $\{x_n\}$ converges to $x$ in the $\ell^{p}$ norm. 
This argument doesn't work for $\ell^\infty$ -- consider the sequence of all ones $u=(1, 1, \ldots) \in \ell^\infty$. For any $y \in Y, \parallel y-u \parallel_\infty \geq 1.$ Therefore no sequence in $Y$ can converge to $u$ in the sup-norm. Indeed, as Martin Sleziak showed above, the closure of $Y$ in $\ell^\infty$ is $c_0$.