Im studying functional analysis and I don't know how to solve this exercise:
We define $T:c_0\rightarrow l^2$ as $T((x_j))=(x_j/j)$. Let $X=c_0$ and $Y=Im(T)$. Let $S\in L(X,Y)$ defined by $S(x)=T(x)$. Show that $S$ is not compact and that is the limit of a sequence of finite range operators.
I don't know how to show that $S$ is not compact and about the second question, in class I have seen that the limit of finite range operators is compact, but I don't know what hypothesis fail in this example.
Can someone help me?
Thanks in advance.