Could we consider a copy of $l^2(G)$ as a subset of $l^2(G,X)$?

Question

Suppose that $G$ is a discrete group, and $X$ is a Banach space. Trivially we have a copy of $C_c(G)$ in $C_c(G,X).$ One may choose an element $x$ in $X$, and fix it. Then for each $f$ in $C_c(G)$, we define $f_x$ to be a function from $G$ to $X$ with $f_x(g)=f(g)x$ for all $g$ in $G$. Then $f_x$ is in $C_c(G,X)$ and we may consider a copy of $C_c(G)$ as a subset of  $C_c(G,X)$. Then if we consider  the completion of $C_c(G,X)$ , could we still have a copy of $l^2(G)$ in this completion( I mean in $l^2(G,X)$)?