If $l^2=\{x=(x_1,x_2,x_3,...):\displaystyle\sum_{i=1}^{\infty}|x_i|^2<\infty\}$ and $T:l^2\rightarrow l^2$ is a linear bounded operator defined by
$T(x_1,x_2,x_3,...)=(0,0,x_1,x_2,x_3,...)$ ,where $x=(x_1,x_2,x_3,...)\in l^2 $
then is $\overline{R(T)}=l^2$ ?
In addition, what is the best way to prove closedness and denseness of subspaces related to bounded operators which are defined on sequence space ?
Here $R(T)=\{(0,0,x_1,x_2,x_3,...)\}$
If $x\in l^2,$ then we have to show either $x\in R(T)$ or $x$ is a limit point of $R(T) ~i.e ~for~ every ~\epsilon>0,~S(x,\epsilon)\cap R(T)\neq\phi. $
I don't how to prove or disprove it. please help.
Thanks.