Let $V$ be a countable set. What is the $l_2(V)$?

Question

I am reading a book about operators in graphs. I know this question may sound obvious to some people, but what is $l_2(V)$ where V is the set of vertices of a graph? Is this like $R^n$ where $n$ is the number of vertices? I want to understand this because then the book talks about bounded linear operators $T: l_2(V)$ → $l_2(V)$. Thank you very much in advance!

Answer 1

Given a countable set $X$, we define $\ell^2(X)=\{(a_x)_{x\in X}\in\mathbb{C}^X:\sum_{x\in X}|a_x|^2<\infty\}$ - this are all the sequences who's sum of square of absolute values converge. Notice that this definition still applies when $X$ is uncountable, but you do have to define what does convergence mean when talking about "uncountable sum".

In general, a not so hard to prove thm is that if $\mathcal{H}$ is a Hilbert space of dimension $\alpha$ and $I$ set of cardinality $\alpha$, then $\mathcal{H}$ is isometrically isomorphic to $\ell^2(I)$. If $X$ is countable (and not finite), this means that $\ell^2(\mathbb{Z})$ is isometrically isomorphic to $\ell^2(X)$ - this is in case you're more comfortable dealing with $\ell^2(\mathbb{Z})$.