What is the connection between the norm of a sequence in $l^2(N)$ and the norm of a function in $L^2(X)$? Little $l$ and big $L$ spaces

Question

If I have a a function in $l^2(N)$, can they converge to a function in $L^2$ of some set $X$ subset of R? I am having a little trouble connecting functions of $l^2$ with finite support with functions in $L^2$?

Answer 1

Let $f \in \ell^2(\Bbb N)$, consider it as a function $f:\Bbb N \to \Bbb C$. We can define an extension $F$ of $f$ to $\Bbb R$ as follows: set $F(x)=f(\lfloor x\rfloor)$ if $x\geq 1$ and $F(x)=0$ if $x<1$. Then $F \in L^2(\Bbb R)$ and we have $\|F\|_{L^2}=\|f\|_{\ell^2}$. The function $\ell^2(\Bbb N) \to L^2(\Bbb R), f \mapsto F$ is an isometric embedding.